Intersection homology with general perversities
Greg Friedman
Texas Christian University
February 15, 2010
To Bruce Williams, for his 60th birthday
Abstract
We study intersection homology with general perversities that assign integers tostratum components with none of the classical constraints of Goresky and MacPherson.We extend Goresky and MacPherson’s axiomatic treatment of Deligne sheaves, and usethese to obtain Poincare and Lefschetz duality results for these general perversities.We also produce versions of both the sheaf-theoretic and the piecewise linear chain-theoretic intersection pairings that carry no restrictions on the input perversities.
2000 Mathematics Subject Classification: Primary: 55N33, 57N80; Secondary:
55N45, 55N30, 57P10
Keywords: intersection homology, perversity, intersection product, Poincare
duality, pseudomanifold, Deligne sheaf
Contents
1 Introduction 2
2 Background 6
2.1 Pseudomanifolds and intersection homology basics . . . . . . . . . . . . . . . 6
2.2 Stratified coefficient systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Efficient perversities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 A generalized Deligne sheaf 13
3.1 Comparison with intermediate extensions . . . . . . . . . . . . . . . . . . . . 18
4 Constructibility and duality 19
4.1 Constructibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Sheaf pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1
5 Intersection pairings on piecewise linear pseudomanifolds 26
5.1 Some basics concerning PL chains and stratified coefficients . . . . . . . . . . 27
5.2 The Goresky-MacPherson pairing . . . . . . . . . . . . . . . . . . . . . . . . 28
5.3 The pairing algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 Introduction
Intersection homology theory is an important tool for the topological study of stratified
spaces, which include algebraic and analytic varieties and certain quotients of manifolds by
group actions. The original motivation for its development was to extend an appropriate
version of Poincare duality to such spaces, and other related properties have followed, in-
cluding versions of the Kahler package for singular varieties. The theory breaks into families
indexed by a parameter, the perversity, which is often limited to a fairly strict range of pos-
sibilities. However, as intersection homology with more general perversities has become an
increasingly indispensable tool, it is imperative to recast as many as possible of the founda-
tional properties of intersection homology theory in this light. We here construct a version
of the Deligne sheaf machinery in this constext, as well as revisiting the geometric PL chain
intersection pairing of Goresky and MacPherson.
More precisely, recall that a perversity parameter for intersection homology is a function
that assigns a number to each stratum (or stratum component) of a stratified pseudomanifold
X. In the original work of Goresky and MacPherson [28], these perversities were assumed
to satisfy very restrictive conditions: A Goresky-MacPherson perversity p assigns the same
number to all stratum components of codimension k, so it can be thought of as a function
p : Z≥2 → Z, and it must satisfy
p(k) ≤ p(k + 1) ≤ p(k) + 1 and p(2) = 0.
These conditions were necessary in order to prove some of the earliest important properties
of intersection homology groups I pH∗(X), including the simultaneous possession of Poincare
duality and topological invariance (independence of the stratification of X).
A number of authors have considered variants of intersection homology that allow more
general notions of perversity, including Beilinson, Bernstein, and Deligne [5]; MacPherson
[40]; King [38]; Cappell and Shaneson [11]; Habegger and Saper [32]; the author [20, 22, 24];
Saralegi [48]; and Hausel, Mazzeo, and Hunsicker [34, 36, 35]. In many of these works, per-
versities are still required to satisfy at least some of the prior conditions, though completely
arbitrary perversities appear as far back as 1982 in the work of Beilinson, Bernstein, and
Deligne on perverse sheaves, and they occur more recently in work of the author [24] and
Saralegi [48]. These general perversities are functions
p : {Connected components of singular strata of X} → Z
without any restrictions whatsoever. These papers also enlarge the class of pseudomanifolds
considered by allowing strata of codimension one. While the topological invariance of in-
tersection homology is lost for these more general perversities, all other standard properties
2
of intersection homology remain true, including Poincare duality, at least with the proper
choice of intersection homology theory.
While general perversities appeared within the realm of perverse sheaves from the begin-
ning, let us describe in a bit more detail some of the recent applications of general perversities
within intersection homology proper:
In [48], Saralegi proved a de Rham theorem for these general perversities, i.e. that inte-
gration induces a linear duality isomorphism between the cohomology of p-perverse liftable
intersection differential forms and chain-theoretic perversity t− p “relative” intersection ho-
mology with real coefficients (see [48] for precise details). Non-traditional perversities also
appear in an analytic setting in the works of Hausel, Hunsicker, and Mazzeo [34, 36, 35],
in which they demonstrate that groups of L2 harmonic forms on a manifold with fibered
boundary can be identified with cohomology spaces associated to intersection cohomology
groups of varying perversities for a canonical compactification of the manifold.
In [24], we used general perversity intersection homology as a critical tool in proving a
Kunneth theorem for intersection homology. We showed that for any two given perversities p,
q on two stratified pseudomanifolds X, Y , there is a family of general perversities (generically
denoted Q) on the product pseudomanifold X × Y such that the intersection chain complex
IQC∗(X × Y ) is quasi-isomorphic to I pC∗(X)⊗ I qC∗(Y ). Even when p and q are Goresky-
MacPherson perversities, Q in general will not be, and so general perversities play a critical
role even in the effort to study more classical forms of intersection homology. Without this
tool, Cohen, Goresky, and Ji [16] were able to obtain only a much more limited intersection
homology Kunneth theorem. Together with James McClure and Scott Wilson, the author
is currently working on a variety of applications of this Kunneth result, including an effort
to understand an intersection cohomology theory founded on cochains and cup products.
Such an effort was not possible previously because the usual front face/back face approach
to cup products (see [44, Section 48]) is not well-behaved on intersection chains. However,
the author’s general perversity Kunneth theorem provides a way to define the cup product
via the Hom dual of the composition
I rHc∗(X)→ IQHc
∗(X ×X)∼=→ I pHc
∗(X)⊗ I qHc∗(X),
where the first map is induced by the geometric diagonal inclusion and the righthand quasi-
isomorphism is the Kunneth theorem of [24]. See [24, 27] for more details.
Thus general perversities have become increasingly useful, and, as such, it is desirable to
have a consistent theory of general perversity intersection homology, unified across the var-
ious existing flavors of intersection homology theory. Recall that Goresky and MacPherson
originally defined intersection homology using simplicial chains on piecewise linear spaces,
but in [29] they formulated an equivalent purely sheaf-theoretic approach. Sheaf theory has
the advantages of applying on more general spaces and of possessing a powerful toolbox,
though perhaps the disadvantage of being further removed from geometric intuition. The
geometric chain approach was extended by King [38] to include singular simplices, as well
as non-Goresky-MacPherson perversities. However, it was shown in [20] that, when con-
sidering King’s perversities, singular chains are not quite the right theory to match with
3
the tools coming from sheaf theory. Thus a modification, the stratified coefficient system,
was introduced in [20], and this is the version of general perversity singular intersection ho-
mology pursued in [20] and [24]; this approach turns out to be equivalent to the “relative”
intersection homology introduced by Saralegi in [48].
Our first goal here is to demonstrate how to generalizes the Deligne sheaf construction
of Goresky-MacPherson [29] in order to provide hypercohomology groups that agree with
the generalized intersection homology of the author and Saralegi. We also generalize the
axiomatic characterization that made the Deligne sheaf so useful in classical intersection
homology and use this to prove duality and pairing results. We further demonstrate that
this generalized Deligne construction is equivalent to a certain intermediate extension in the
language of perverse sheaves.
At the other end of the spectrum, we expand also the geometric ramifications of general
perversities by extending some of the earliest Goresky-MacPherson results in the realm of
piecewise-linear chain complexes by extending the PL intersection pairing to arbitrary pairs
of perversities.
While some of our sheaf theoretic results may be implicit (or, in some cases, explicit)
in developments from the category perverse sheaves, such an approach involves employment
of considerably more abstract categorical formalisms. We hope that our more geometrically
explicit constructions will be more approachable and better adapted to future applications.
We also hope that, ultimately, having several approaches to general perversity intersection
homology, coming with trade-offs between their varying levels of geometrical explicitness and
their abstract generality, will provide scientific utility.
We proceed as follows.
Section 2 contains background and notation.
In Section 3, we observe that the classical Deligne sheaf construction is insufficient to pro-
vide a complete sheaf-theoretic approach to intersection homology with general perversities.
To remedy this problem, we introduce a generalization of the sheaf truncation functor to re-
place the classical sheaf truncation functor in the Deligne construction. We then show that
general perversity intersection homology possesses an axiomatic characterization founded
upon the properties of this generalized Deligne sheaf.
In Section 4, we show that our generalized Deligne sheaves are constructible and that
this implies a general perversity version of the Poincare-Verdier-Goresky-MacPherson duality
theorem on stratified pseudomanifolds. The full statement of the theorem can be found
below as Theorem 4.3. We here state the special case over an orientable pseudomanifold
without boundary and for a constant coefficient sheaf whose stalks are the field F . In
the statement, Q∗ denotes our generalized Deligne sheaf, D denotes the Verdier dualizing
functor, U1 = X −Xn−1, and t is the top perversity, t(Z) = codim(Z)− 2.
Theorem 1.1. Let X be an orientable n-dimensional stratified pseudomanifold, and let p
and q be general perversities such that p + q = t. Then DXQ∗p[−n] ∼= Q∗q(DU1F [−n]) in the
derived category of sheaves on X.
This implies a more familiar-looking statement, which forms the third item of Corollary
4.4:
4
Corollary 1.2. If X is closed and orientable and p, q are general perversities with p+ q = t,
then
I qHn−i(X; Q0) ∼= Hom(I pHi(X; Q0),Q).
This includes the original duality result of Goresky and MacPherson as a special case.
The subscript 0 on the Q coefficients reflects the use of a stratified coefficient system; see
Section 2.
While the preceding statements might be somewhat expected considering the theorems
of Goresky and MacPherson, their generalization to our current setting does have some
nice consequences, including a simple proof of the following Lefschetz duality theorem for
pseudomanifolds with boundary.
Corollary 1.3. If X is a compact and orientable stratified pseudomanifold with boundary
and p, q are general perversities with p+ q = t, then
I qHn−i(X; Q0) ∼= Hom(I pHi(X, ∂X; Q0),Q).
This corollary follows from Corollary 4.4 and the discussion of Lefschetz duality that
follows it in Section 4.1
In Section 4.3, we explore the sheaf-theoretic intersection homology pairings. The fol-
lowing theorem appears below as Theorem 4.6 and is much more general than the classical
result (see [29, 7]), in which no pairing morphism m can exist at all if p+ q 6≤ t.
Theorem 1.4. Given a pairing of local systems m1 : E ⊗ F → G on X −Xn−1 and general
perversities such that p(Z) + q(Z) ≤ r(Z) for all singular strata Z, then in the bounded
derived category Db(X), there is a unique morphism m : Q∗p(E∗)L⊗ Q∗q(F) → Q∗r(G) that
restricts to m1 on X − Xn−1. Furthermore, if r ≤ t and G = O, the orientation sheaf of
X −Xn−1, then there exists a pairing m : Q∗p(E∗)L⊗Q∗q(F) → D∗X [−n] that restricts to m1
on X −Xn−1, where D∗X [−n] is the shift of the Verdier dualizing complex on X.
Finally, in Section 5, we return to the geometry of simplicial chains on PL spaces. In
Theorem 5.4 we provide a generalization of the original Goresky-MacPherson intersection
pairing. We state here a nice special case, assuming X is compact and orientable:
1 This intersection homology Lefschetz duality is currently being utilized by the author and Hunsickerin their study of intersection homology versions of Novikov additivity and Wall non-additivity for perversesignatures [26]. These are the signatures of the nondegenerate intersection pairing on im(I pH2n(X4n) →I qH2n(X4n, ∂X4n)), when p ≤ q; the well-known Witt-space signature is a special case. Signatures arisingthrough intersection homology and the closely related analytic L2-cohomology and L2 Hodge theory havebeen the subject of intense study since the beginnings of the subject. They have been studied topologicallyby researchers including Goresky and MacPherson [28, 29], Siegel [49], and various combinations of Banagl,Cappell, Libgober, Maxim, Shaneson, and Weinberger, whose papers on the topic include, among manyothers, [3, 1, 4, 8, 10, 9]. There has also been much interest in the analytic study of these intersectionhomology signatures as they arise in L2-cohomology and L2 Hodge theory and as they may relate to dualityin string theory, such as through Sen’s conjecture on the dimension of spaces of self-dual harmonic forms onmonopole moduli spaces. Results in these areas and closely related topics include those of Muller [43]; Dai[17]; Cheeger and Dai [15]; Hausel, Hunsicker, and Mazzeo [34, 36, 35]; Saper [46, 45]; Saper and Stern [47];and Carron [12, 14, 13]; and work on analytic symmetric signatures is currently being pursued by Albin,Leichtmann, Mazzeo and Piazza.
5
Theorem 1.5. Suppose x ∈ I pCi(X; Z0), y ∈ I qCj(X; Z0) are such that the pairs (|x|, |y|),
(|∂x|, |y|), and (|x|, |∂y|) are in stratified general position. Then there is a well-defined in-
tersection x t y ∈ I rCi+j−n(X; Z0) for any r ≥ p+ q.
Once again, such a general pairing does not exist in classical intersection homology,
in which it is not possible to intersect a p-allowable chain and a q-allowable chain unless
p+ q ≤ t.
We also show in Section 5 that the results of [25] on the existence of partial DGA
structures on intersection chain complexes generalize to include general perversities.
Some of the arguments we present here adhere in general form to well-known paths
in intersection homology theory or to our other recent work. However, there are several
novelties that require special attention and detailed work. These include the introduction
of our generalized sheaf truncation functor, as well as the details of the PL chain pairing,
for which it is necessary to work with what can be best described as relative homology
analogues of more classical “absolute homology” arguments. In our exposition, we hope
to have found a middle road that does not repeat too much that can be found elsewhere
in the literature but that is sufficiently detailed to allow the reader to appreciate the new
results and modified techniques that arise in the study and application of general perversity
intersection homology.
Remark 1.6. A detailed expository survey of intersection homology with general perversities,
including an overview of some of the present results, can be found in [19].
Acknowledgment. I sincerely thank an anonymous referee for pointing out the appear-
ance of general perversities in the theory of perverse sheaves.
2 Background
We begin with a brief review of definitions, referring the reader to sources such as [28, 29,
7, 39, 2, 38, 22, 19] for more thorough background. We encourage the experts also to skim
this section, as we allow a few unconventional generalizations.
2.1 Pseudomanifolds and intersection homology basics
Pseudomanifolds. Let c(Z) denote the open cone on the space Z, and let c(∅) be a point.
A stratified paracompact Hausdorff space Y (see [29] or [11]) is defined by a filtration
Y = Y n ⊃ Y n−1 ⊃ Y n−2 ⊃ · · · ⊃ Y 0 ⊃ Y −1 = ∅
such that for each point y ∈ Y i−Y i−1, there exists a distinguished neighborhood U of y such
that there is a compact Hausdorff space L, a filtration of L
L = Ln−i−1 ⊃ · · · ⊃ L0 ⊃ L−1 = ∅,
6
and a homeomorphism
φ : Ri × c(L)→ U
that takes Ri× c(Lj−1) onto Y i+j ∩U . The Y i are called skeleta. We denote Yi = Y i−Y i−1;
this is an i-manifold that may be empty. We refer to the connected components of the
various Yi as strata2. If a stratum Z is a subset of Yn = Y − Y n−1 it is called a regular
stratum; otherwise it is called a singular stratum. L is called a link. The depth of a stratified
space is the number of distinct skeleta it possesses minus one.
Usually, a stratified (topological) pseudomanifold of dimension n is defined to be a strat-
ified paracompact Hausdorff space X such that Xn−1 = Xn−2, X − Xn−2 is a manifold
of dimension n dense in X, and each link L is, inductively, a stratified pseudomanifold;
a space is a (topological) pseudomanifold if it can be given the structure of a stratified
pseudomanifold for some choice of filtration. In this paper, we will also allow for the possi-
bility that Xn−1 6= Xn−2. When we do assume Xn−1 = Xn−2, intersection homology with
Goresky-MacPherson perversities is known to be a topological invariant; in particular, it
is invariant under choice of stratification (see [29], [7], [38]). Examples of pseudomanifolds
include complex algebraic and analytic varieties (see [7, Section IV]).
A stratified pseudomanifold is orientable (respectively oriented) if X −Xn−2 is.
We refer to the link L in the distinguished neighborhood U of y as the link of y or of
the stratum containing y; it is, in general, not uniquely determined up to homeomorphism,
though if X is a pseudomanifold it is unique up to, for example, stratum preserving homotopy
equivalence (see, e.g., [21]), which is sufficient for the intersection homology type of the link
of a stratum to be determined uniquely. Thus there is no harm, in general, of referring to
“the link” of a stratum instead of “a link” of a stratum.
A piecewise linear (PL) pseudomanifold is a pseudomanifold with a PL structure compat-
ible with the filtration, meaning that each skeleton is a PL subspace, and such that each link
is a PL pseudomanifold and the distinguished neighborhood homeomorphisms U ∼= Rn−k×cLare PL homeomorphisms.
We will assume all pseudomanifolds X to have a fixed given stratification X.
Intersection homology. We will work mostly with singular chain intersection homology
theory, which was introduced in [38] with finite chains (compact supports) and generalized
in [22] to include locally-finite but infinite chains (closed supports). Singular intersection
homology can be defined on any filtered space, but we limit ourselves to stratified pseudo-
manifolds.
A Goresky-MacPherson perversity (or GM perversity) p is a function p : Z≥2 → Z such
that p(k) ≤ p(k + 1) ≤ p(k) + 1 and p(2) = 0. The elements of Z≥2 correspond to the sets
Xk. Here the domain is Z≥2 because Goresky and MacPherson did not allow codimension
1 strata. King [38] introduced loose perversities, which are completely arbitrary functions
p : Z≥0 → Z. We will go even further, following Saralegi [48], and define a general perversity
2This definition agrees with some sources, but is slightly different from others, including our own pastwork, which would refer to Yi as the stratum and what we call strata as “stratum components.”
7
on a stratified pseudomanifold X to be any function3 p : {singular strata of X} → Z.
Given a stratified pseudomanifold X, a general perversity p, and an abelian group G,
one defines the intersection chain complex I pCc∗(X;G) as a subcomplex of Cc
∗(X;G), the
complex of compactly supported singular chains on X, as follows: An i-simplex σ : ∆i → X
in Cci (X) is allowable if
σ−1(Z) ⊂ {i− codim(Z) + p(Z) skeleton of ∆i}
for any singular stratum Z of X. The chain ξ ∈ Cci (X;G) is allowable if each simplex
with non-zero coefficient in ξ or in ∂ξ is allowable. I pCc∗(X;G) is the complex of allowable
chains. I pC∞∗ (X;G) is defined similarly as the complex of allowable chains in C∞∗ (X;G),
the complex of locally-finite singular chains. Chains in C∞∗ (X;G) may be composed of an
infinite number of simplices (with their coefficients), but for each such chain ξ, each point in
X must have a neighborhood that intersects only a finite number of simplices (with non-zero
coefficients) in ξ. See [22] for more details.
The associated homology theories are denoted I pHc∗(X;G) and I pH∞∗ (X;G) and called
intersection homology with, respectively, compact or closed supports.
N.B. We will often omit the decorations c or ∞ when these theories are equivalent or
when our statements apply to either case.
Relative intersection homology is defined similarly, in the obvious way, though we note
that the filtration on a subspace will always be that inherited from the larger space by
restriction, and, in the closed support case, all chains are required to be locally-finite in the
larger space.
If p is a GM perversity and X has no strata of codimension one, then it is also possible
to define intersection chains with coefficients in a local system of groups G that is defined
only on X −Xn−2. See [29, 22, 19] for more details.
Sheaves. Recall from [22] that one can define a sheaf complex4 I pS∗ on the n-dimensional
stratified pseudomanifold X as the sheafification of the presheaf U → I pC∞n−∗(X,X − U ;G)
or, equivalently, of the presheaf U → I pCcn−∗(X,X − U ;G). It is shown in [22] that the for-
mer presheaf is conjunctive for coverings and has no non-trivial global sections with empty
support. Furthermore, the sheaf I pS∗ is homotopically fine. As a consequence, the hyper-
cohomology Hi(X; I pS∗) is isomorphic to I pH∞n−i(X;G), and Hic(X; I pS∗) ∼= I pHc
n−i(X;G).
By [29], if p is a GM perversity and X is a pseudomanifold, then I pS∗ is quasi-isomorphic
to the Deligne sheaf, and hence they are interchangeable in the derived category of sheaves.
3 Technically, our definition of a general perversity is not strictly more general than King’s because heallows loose perversities to be defined on regular strata and takes this into account in defining intersectionchains. In practice, however, if Z is a regular stratum, the only meaningful distinction for intersectionhomology is between the cases p(Z) ≥ 0 and p(Z) ≤ −1. The latter condition would force chains to avoidthe regular stratum altogether, which is usually not worth considering (see the discussion of this point in[19, Remark 6.1]), and it makes some later statements technically simpler to avoid imposing repeatedly thecondition p(Z) ≥ 0 for regular strata. The most elegant solution, used by Saralegi in [48], seems to be todefine p only on the singular strata and to build this restriction into the definition of intersection chains.
4We tend to leave the coefficients out of the sheaf notation for simplicity; the coefficients will always beknown from context.
8
To define the Deligne sheaf, let Uk = X − Xn−k, and let ik : Uk ↪→ Uk+1 be the inclusion.
Rik∗ denotes right derived functor of the push forward ik∗ and τ≤m is the truncation functor
that acts on the complex A∗ (in any appropriate category) as
(τ≤mA∗)i =
0, i > m,
ker(d : Am → Am+1), i = m,
Ai, i ≤ m.
Recall that H i(τ≤mA∗) = 0 for i > m, and H i(τ≤mA
∗) = H i(A∗) for i ≤ m. Let GU1 be the
constant sheaf G on the open manifold U1. Then the Deligne sheaf P∗p is defined inductively
as τ≤p(n)Rin∗ . . . τ≤p(1)Ri1∗GU1 .
It is again possible to begin with a local system of groups G that is defined only on
X −Xn−1; one simply replaces GU1 with G.
2.2 Stratified coefficient systems
Motivation. When working with perversities p for which p(Z) > codim(Z) − 2 for some
stratum Z (we call such perversities superperversities5 ), it is useful to make a technical
adjustment to the definition given above of chain-theoretic intersection homology. This
adjustment was introduced in [22] under the guise of using a “stratified coefficient system” G0
and independently by Saralegi [48] in the form of a certain relative intersection homology; it is
shown in [24] that the two approaches yield isomorphic chain complexes. The motivation for
stratified coefficients in [22] was the desire to construct a singular chain version of intersection
homology with superperversities that agrees with the Deligne sheaf version from [11], while
Saralegi’s purpose was to prove a de Rham theorem for general perversities. An added
benefit is that stratified coefficients let us start with a local coefficient system on X −Xn−1
for any general perversity, which would not otherwise be possible.
Before providing the definition, we note that the main technical point necessitating strat-
ified coefficients is the need to generalize the intersection homology cone formula so as to
obtain the following proposition. This proposition combines [22, Proposition 2.18] with the
isomorphism I pHci (cL, L×R;G0) ∼= I pH∞i (cL;G0), which follows from [22, Lemma 2.12]. In
[22], these propositions are not proven in quite the generality stated here, but the proofs
generalize immediately; see also [19].
Proposition 2.1. Let L be an n − 1 dimensional filtered space with stratified coefficient
system G0. Let v be the vertex of cL, let cL be filtered such that (cL)0 = v and (cL)i =
c(Li−1)− v for i > 0. Then6
I pHci (cL;G0) ∼=
{0, i ≥ n− 1− p({v}),I pHc
i (L;G0), i < n− 1− p({v}).5The term “superpervisity” was first used by the author in [22, 20] to refer to the perversities considered
by Cappell and Shaneson in [11].6On the second line, we should technically use I pHi−1(L;G0|L), but we will leave such restrictions tacit
throughout.
9
If L is compact, then
I pHci (cL, L× R;G0) ∼= I pH∞i (cL;G0) ∼=
{I pHi−1(L;G0), i ≥ n− p({v}),0, i < n− p({v}).
This proposition is familiar for GM-perversity intersection homology, but it would not
be true for general perversities without the stratified coefficient systems. The reader should
compare with [38, Proposition 5]; the small but important difference is entirely in degree 0.
While this seems like a minor point, this proposition in its usual form for GM perversities
is a key player in proving all of the major results of the theory, and this continues to be borne
out for general perversities in [22, 24, 48] and in the results that follow. This seems ample
evidence that stratified coefficient systems (or, equivalently, Saralegi’s relative groups) are
worth using. Furthermore, when p is a Goresky-MacPherson perversity (or, more generally,
if p(Z) ≤ codim(Z) − 2 for each singular stratum Z), then I pC∗(X;G0) ∼= I pC∗(X;G) as
observed in [20, Proposition 2.1]7. Thus intersection chains with stratified coefficients are a
legitimate extension of the traditional setting.
Remark 2.2. The intersection homology sheaves constructed by Habegger and Saper [32]
reflect singular intersection homology for which p(Z) ≥ codim(Z)−1 may occur but stratified
coefficients are not used. Thus the hypercohomology of their sheaves is not the same as the
superperverse intersection homology found in [11, 22, 48]. Habegger and Saper also continue
to require that p(k) ≤ p(k + 1) ≤ p(k) + 1. They find a duality theorem in this setting, but
it is technically more complicated than the Goresky-MacPherson duality and that produced
here. See [32, 19] for more details.
Definition of stratified coefficients. Suppose given a filtered space X and local system
of coefficients G on X −Xn−1. Then the stratified coefficient system G0 is defined to consist
of the pair of coefficient systems given by G on X−Xn−1 and the constant 0 system on Xn−1,
i.e. we think of G0 as consisting of a locally constant fiber bundle GX−Xn−1 over X −Xn−1
with fiber G (with the discrete topology) together with the trivial bundle on Xn−1 with stalk
0. Then a coefficient n of a singular simplex σ can be described by a lift of σ|σ−1(X−Xn−1) to
GX−Xn−1 together with the trivial “lift” of σ|σ−1(Xn−1) to the 0 system on Xn−1. A coefficient
of a simplex σ is considered to be the 0 coefficient if it maps each point of ∆ to the 0 section
of one of the coefficient systems. If nσ is a simplex σ with its coefficient n, its boundary is
given by the usual formula ∂(nσ) =∑
j(−1)j(n ◦ ij)(σ ◦ ij), where ij : ∆i−1 → ∆i is the jth
face inclusion map. In this way we can form a chain complex C∗(X;G0).
The basic idea behind the definition is essentially that when we consider allowability of
chains with respect to a perversity, simplices with support entirely in Xn−1 should vanish and
thus not be counted for allowability considerations. Thus the intersection homology chain
complexes I pCc∗(X;G0) and I pC∞∗ (X;G0) are defined just as I pCc
∗(X;G) and I pC∞∗ (X;G)
are, but replacing the coefficients of simplices with coefficients in G0. Allowability of a
simplex is thus determined as above, but all simplices with support in Xn−1 vanish due
7The proposition there has the hypotheses that Xn−1 = Xn−2, but this hypothesis in not necessary iffor each Z ⊂ Xn−1, p(Z) ≤ codim(Z)− 2 = −1, a condition which was not allowed in [20].
10
to the 0 coefficient they must carry, and so they are automatically allowable. This yields
nothing new when considering the allowability of the i-simplices of an i-chain ξ, but now
any simplices of ∂ξ with support in Xn−1 vanish, which may come into play in determining
the allowability of ξ as a chain. More details can be found in [22].
Relationship with Saralegi’s relative chains. Let G be a constant coefficient sys-
tem. As noted in [22], use of the coefficient system G0, in which Xn−1 carries a formal
0 coefficient system, is not the same as attempting to take relative intersection homology
I pH∗(X,Xn−1;G). For one thing, if p(Z) ≤ codim(Z)− 1, there is no such chain submodule
as I pC∗(Xn−1;G) because no allowable chains are contained entirely within Xn−1. However,
Saralegi’s idea in [48] for an intersection chain complex satisfying Proposition 2.1 is a correct
refinement of this idea. He defined a chain complex S pC∗(X;Xt−p) as
S pCc∗(X,Xt−p;G) =
(ApC∗(X;G) + AC p+1∗ (Xt−p;G)) ∩ ∂−1
(ApC∗−1(X;G) + AC p+1
∗−1 (Xt−p;G))
AC p+1∗ (Xt−p;G) ∩ ∂−1AC p+1
∗−1 (Xt−p;G),
where t is the top perversity, t(Z) = codim(Z)− 2, ApCi(X) is generated by the p-allowable
i-simplices of X, Xt−p is the closure of the union of the singular strata Z of X such that
t(Z)− p(Z) < 0, and Ap+1Ci(Xt−p) is generated by the t− p− 1 allowable i-simplices with
support in Xt−p. Once again, we see that the idea is to have a complex made up of allowable
chains but to kill simplices that lie within the strata where the perversity is too high.
In [24], we proved that I pCc∗(X;G0) is quasi-isomorphic to S pCc
∗(X,Xt−p;G). More
precisely, the proof there uses the extra assumption that the coefficient of any simplex in a
chain of I pC∗(X;G0) must lift to the same coefficient at all points of σ−1(X −Xn−1), and
with this assumption, I pCc∗(X;G0) and S pCc
∗(X,Xt−p;G) are actually isomorphic. However,
it is also noted in [24] that this variant of the definition of I pCc∗(X;G0) yields a chain complex
that is quasi-isomorphic with the chain complex we have been using here. Furthermore, this
extra assumption will automatically be true if p is an “efficient perversity,” a notion that is
defined in the next subsection, where we also show that p can always be replaced with an
efficient perversity without changing the intersection homology groups.
Throughout this paper, we will use the stratified coefficient formulation, which we think
is slightly simpler to use in most of our arguments, largely because it avoids quotient groups.
Also, it is not quite clear how to extend Saralegi’s approach to local coefficient systems.
Basic Properties. As shown in [22], even with general perversities and G0 coefficients,
many of the basic properties of I pH∗(X;G0) established in [38] and [22] hold with little or
no change to the proofs, such as stratum-preserving homotopy equivalence, excision, the
Kunneth theorem for which one term is an unstratified manifold, Mayer-Vietoris sequences,
etc.
It also remains true with general perversities and stratified coefficients that we can define
a sheaf I pS∗ as the sheafification of the presheaf U → I pC∞n−∗(X,X− U ;G0) or, equivalently,
of the presheaf U → I pCcn−∗(X,X − U ;G0). Again, the former presheaf is conjunctive for
coverings and has no non-trivial global sections with empty support, and the sheaf I pS∗ is
11
homotopically fine. Thus, the hypercohomology Hi(X; I pS∗) is isomorphic to I pH∞n−i(X;G0),
and Hic(X; I pS∗) ∼= I pHc
n−i(X;G0). For simplicity of notation, we generally write I pS∗,without noting explicitly the coefficients.
N.B. Throughout this paper, we will always assume stratified coefficients are in use unless
explicitly noted otherwise.
2.3 Efficient perversities
In this section, we show that many perversities provide redundant information. In other
words, there is no need to consider perversities that are too general.
Definition 2.3. We will say that a general perversity p is efficient if, for each singular
stratum Z, −1 ≤ p(Z) ≤ codim(Z)− 1.
Given a general perversity p, we can associate an efficient perversity p as follows: Define
p by
p(Z) =
codim(Z)− 1, if p(Z) ≥ codim(Z)− 1,
p(Z), if 0 ≤ p(Z) ≤ codim(Z)− 2,
−1, if p(Z) ≤ −1.
We now show that we can effectively eliminate consideration of perversities that are not
efficient when studying intersection homology.
Lemma 2.4. Let p be a general perversity and X a pseudomanifold, possibly with codi-
mension one strata. Let p be the associated efficient perversity. Then I pH∗(X;G0) ∼=I pH∗(X;G0).
Corollary 2.5. Let X− denote the union of the singular strata Z of X such that p(Z) ≤ −1.
Then I pH∗(X;G0) ∼= I pH∗(X −X−;G0).
Proof. This follows directly from the lemma and from the definition of the intersection chain
complex by replacing p with a perversity such that p(Z)� 0 for each Z ⊂ X−.
Proof of Lemma 2.4. We proceed in two steps. First, let p be the perversity defined so that
p(Z) = −1 if p(Z) ≤ −1 and p(Z) = p(Z) otherwise. Then there is a natural inclusion
I pC∗(X;G0) ↪→ I pC∗(X;G0) (for either closed or compact supports). This induces a map
of sheaves I pS∗ → I pS∗, which we claim is a quasi-isomorphism. We induct on depth,
noting that we clearly have a quasi-isomorphism on depth 0 pseudomanifolds, which have
no singular strata. So suppose that we have shown the claim through depth j − 1 and
that x ∈ X, where X has depth j. It suffices to compute locally, and by the standard
computations (e.g. [38, 22]),
Hn−∗(I pS∗x) ∼= I pH∞∗ (Rn−k × cL;G0)∼= I pH∞∗−n−k(cL;G0),
and similarly for p. In fact, the map of cohomology stalks of sheaves comes down to the
map induced by inclusion I pH∞∗ (cL;G0)→ I pH∞∗ (cL;G0). But by the cone formula and the
12
induction hypothesis, since depth(L) < j, this is an isomorphism: if p(Z) ≤ −1, each of these
groups is 0 for all ∗, using that cL is k dimensional and so IH∗(cL;G0) vanishes for ∗ > k
automatically (an argument for this can be found below within the proof of Proposition
3.7). If p(Z) ≥ 0, then for i ≥ k − p(Z) = k − p(Z), I pH∞i (cL;G0) ∼= I pHi−1(L;G0) ∼=I pHi−1(L;G0) ∼= I pH∞i (cL;G0), where the middle isomorphism follows from the induction
hypothesis.
Similarly, using the “opposite extreme” of the cone formula, there is a quasi-isomorphism
I pC∗(X;G0) ↪→ I pC∗(X;G0), where p(Z) = p(Z) when p(Z) ≤ codim(Z) − 2 and p(X) =
codim(Z)− 1 otherwise.
The lemma now follows by noting that p can be obtained from p by first forming p and
then applying the p construction to p or vice versa.
Lemma 2.6. If p is an efficient perversity, than so is t − p, where t is the top perversity,
t(Z) = codim(Z)− 2.
Proof. Since −1 ≤ p(Z) ≤ codim(Z)− 1, t(Z) + 1 ≥ t(Z)− p(Z) ≥ t(Z)− codim(Z) + 1, or
codim(Z)− 1 ≥ t(Z)− p(Z) ≥ −1.
Efficient perversities and interiors of simplices. Efficient perversities have a nice
feature that makes them technically better behaved than the more general perversities. If p
is a perversity for which p(Z) ≥ codim(Z) for some singular stratum Z, then any i-simplex
σ will be p-allowable with respect to Z. In particular, Z will be allowed to intersect the
image under σ of the interior of ∆i. As such, σ−1(X − Xn−1) could potentially have an
infinite number of connected components, and a coefficient of σ might lift each component
to a different branch of G, even if G is a constant system. This could potentially lead to
some pathologies, especially when considering intersection chains from the sheaf point of
view. However, if p is efficient, then for a p-allowable σ we must have σ−1(X−Xn−1) within
the i−1 skeleton of ∆i. Hence assigning a coefficient lift value to one point of the interior of
∆i determines the coefficient value at all points (on σ−1(X −Xn−1) by the unique extension
of the lift and on σ−1(Xn−1), where it is 0). This is technically much simpler and makes the
complex of chains in some sense smaller.
In [24], the complex I pC∗(X;G0) was defined with the assumption that this “unique
coefficient” property holds, meaning that a coefficient should be determined by its lift at a
single point. However, as noted in [24, Appendix], even for inefficient perversities, this does
not change the intersection homology. Having introduced efficient perversities here, we are
instead free to assume all perversities are efficient, without loss of any information (at least
at the level of quasi-isomorphism), and this provides a reasonable way to avoid the issue
entirely.
3 A generalized Deligne sheaf
We now turn to sheaf theoretic versions of general perversity intersection homology.
13
The need for a generalization of the Deligne construction. We first provide an
example showing that general perversity intersection homology is not necessarily given by
hypercohomology of the classical Deligne sheaf, even if p(Z) = p(Z ′) when codim(Z) =
codim(Z ′).
Proposition 3.1. Let p be a general perversity and X an n-dimensional pseudomanifold.
Then neither I pH∗(X; Z) nor I pH∗(X; Z0) is necessarily isomorphic to Hn−∗(X;P∗), where
P∗ is the perversity-p Deligne sheaf.
Proof. We provide an example. We will use the more traditional notation for perversities
taking codimensions as inputs.
Let X = S2T 2, the doubly suspended torus, stratified as such (so that X0 consists of
two points, X1 − X0 consists of two arcs, and X3 = X2 = X1), and let p be the general
perversity given by p(0) = p(1) = p(2) = 0, p(3) = 3, p(4) = 0. We first compute I pH∗(X)
from the singular chain definition of intersection homology.
Since ST 2 has dimension 3 and p(X0) = 0, we have by the intersection homology sus-
pension formula (see [38, Proposition 5]),
I pH∗(X) ∼=
I pH∗−1(ST 2), ∗ > 3,
0, ∗ = 3− p(4) = 3,
I pH∗(ST2), ∗ < 3,
and, using Proposition 2.1 and a Mayer-Vietoris sequence, the same formula holds with Zreplaced by Z0.
For I pH∗(ST2), since p(X1) = 3, all chains are allowable, and
I pH∗(ST2; Z) ∼= H∗(ST
2) ∼=
Z, ∗ = 3,
Z2, ∗ = 2,
0, ∗ = 1,
Z ∗ = 0.
If we replace the Z coefficients with Z0 coefficients, the only change is to replace the Z in
degree 0 with a 0.
Putting these together, we have
I pH∗(S2T 2) ∼= H∗(S
2T 2) ∼=
Z, ∗ = 4,
0, ∗ = 3,
Z2, ∗ = 2,
0, ∗ = 1,
Z, ∗ = 0,
and, again, if we replace the Z coefficients with Z0 coefficients, the only change is to replace
the Z in degree 0 with a 0.
We claim, on the other hand, that H∗(X;P∗) ∼= H∗(X; Z). Thus
14
H4−∗(X;P∗) ∼= H4−∗(X) ∼=
Z, ∗ = 4,
0, ∗ = 3,
0, ∗ = 2,
Z2, ∗ = 1,
Z, ∗ = 0,
by the standard suspension formula for cohomology. This provides the desired contradiction.
To verify the claim, we first show that P∗ ∼= τ≤0Rj∗ZX4 , where j = i4 ◦ i3 and ∼= denotes
quasi-isomorphism (since we work in the derived category of sheaf complexes). Recall that,
by definition, P∗ = τ≤0Ri4∗τ≤3Ri3∗ZX4 . Now if x ∈ X1 = X4−3, then due to the pushforward,
the cohomology of the stalk of Ri3∗ZX4 is that of the link T 2 (since H∗((Ri3∗ZX4)x)∼=
lim−→x∈U H∗(U − U ∩X1; Z) ∼= H∗(U − U ∩X1; Z) ∼= H∗(T 2), where for the last isomorphism
we have used that U − U ∩X1 is homotopy equivalent to T 2 since U can be assumed to be
a distinguished neighborhood R1 × cT 2). Thus the truncation τ≤3 occurs at a sufficiently
large dimension that τ≤3Ri3∗ZX4 is quasi-isomorphic to Ri3∗ZX4 . Then τ≤0Ri4∗τ≤3Ri3∗ZX4∼=
τ≤0Ri4∗Ri3∗ZX4∼= τ≤0Rj∗ZX4 .
Now, we notice that X can also be stratified as a pseudomanifold with two strata X ⊃ X1,
where X1 = X1 ∪X0 ∼= S1, since the double suspension of a space is the same thing as the
join of that space with S1. Using this stratification, τ≤0Rj∗ZX4 is the perversity-0 Deligne
sheaf, so H∗(X; τ≤0Rj∗ZX4)∼= I 0H4−∗(X). Since X is a normal pseudomanifold, this is just
H∗(X; Z) (see [33, Section I.4.1]).
A Deligne sheaf for general perversities. In this section, we define a generalization of
the sheaf truncation functor that, when substituted into the Deligne sheaf formula, yields
a sheaf quasi-isomorphic to I pS∗ even if p is a general perversity. This truncation functor
is a further generalization of the “truncations over a closed subset” functor presented in
[29, Section 1.14] and attributed to Deligne; that functor is used in [29, Section 9] to study
extensions of Verdier dual pairings in the context of classical perversity intersection homology.
Definition 3.2. Let A∗ be a sheaf complex on X, and let F be a locally-finite collection of
subsets of X. Let |F| = ∪V ∈FV . Let P be a function F→ Z. Define the presheaf T F≤PA∗ as
follows. If U is an open set of X, then we let
T F≤PA
∗(U) =
{Γ(U ;A∗), U ∩ |F| = ∅,Γ(U ; τ≤inf{P (V )|V ∈F,U∩V 6=∅}A∗), U ∩ |F| 6= ∅.
Restriction is well-defined because if m < n there is a natural inclusion τ≤mA∗ ↪→ τ≤nA∗.Let the generalized truncation sheaf τF
≤PA∗ be the sheafification of T F≤PA∗.
For maps f : A∗ → B∗ of sheaf complexes over X, we can define τF≤Pf in the obvious way.
In fact, T F≤Pf is well-defined by applying the ordinary truncation functors on the appropriate
subsets, and we obtain τF≤Pf again by passing to limits in the sheafification process.
15
The following lemma contains the key facts we will need about the generalized truncation;
they all follow immediately from the definition and the properties of the usual truncation
τ≤m.
Lemma 3.3. 1. τF≤P is a functor of sheaf complexes on X.
2. There is an inclusion of sheaves τF≤PA∗ ↪→ A∗.
3. Suppose F has the property that for each V ∈ F and each x ∈ V , there is a neighborhood
U of x such that U ∩ V ′ = ∅ for each V ′ ∈ F such that V ′ 6= V . Then (τF≤PA∗)|X−|F| =
A∗|X−|F| and for each V ∈ F , (τF≤PA∗)|V = (τ≤P (V )A∗)|V = τ≤P (A∗|V ).
Remark 3.4. It follows from the last statement of the lemma that if F = {X}, then τF≤PA∗ =
τ≤P (X)A∗, which is a truncation in the usual sense.
Remark 3.5. T F≤PA∗ will not necessarily be a sheaf, so the sheafification in the definition is
necessary. For example, let X = {v, w} be the two point discrete set, and let each stalk
of A∗ be the the chain complex that is trivial except for the isomorphism Z → Z from
degree 0 to degree 1. Now, let F = {{w}} and let P ({w}) = 0. Consider T F≤PA∗. We have
T F≤PA∗({v}) = A∗({v}), but T F
≤PA∗({w}) = T F≤PA∗(X) = 0, since, on each of the latter sets,
the kernel of d : A0 → A1 is trivial. But this means that the trivial section of T F≤PA∗ over
{w} and the section that is 1 in degree 0 over {v} cannot be pieced together into a global
presheaf section. So T F≤PA∗ is not conjunctive and therefore not a sheaf.
Now, let X be an n-pseudomanifold, and recall the standard notation: We let Xk be
the k-skeleton, Xk = Xk − Xk−1, Uk = X − Xn−k, and ik : Uk ↪→ Uk+1. Notice that
Uk+1 = Uk ∪ Xn−k. Also, we will write τXk≤P , allowing Xk to stand for the set of connected
components of Xk. We continue to allow the possibility that our pseudomanifolds possesses
a codimension 1 stratum.
Given a fixed general perversity p and a local coefficient system G on X −Xn−1, let
Q∗p(G) = τX0≤pRin∗ . . . τ
Xn−1
≤p Ri1∗G.
This is the generalization of the Deligne sheaf using our generalized truncations instead of
the usual truncations. We will often omit G from the notation and refer only to Q∗p or Q∗when the context is sufficient. We can now state and prove our main theorem of this section:
Theorem 3.6. Given a general perversity p and a system of coefficients G on U1 = X−Xn−1,
Q∗p(G0⊗O0) is quasi-isomorphic to I pS∗(G0), where O is the orientation sheaf on X−Xn−1
and I pS∗(G0) is the sheaf of p-perversity intersection chains with coefficients in G0.
The proof consists of two propositions. The first tells us that I pS∗ satisfies the Deligne
sheaf axioms with the perversity being replaced by a general perversity; the second tells us
that any sheaf satisfying the axioms is quasi-isomorphic to the generalized Deligne sheaf.
The first proposition is mostly analogous to the standard case. We consider the following
axioms (AX1)p,X,G for a sheaf S∗, where X refers to the fixed stratification on X:
16
1. S∗ is bounded, S i = 0 for i < 0, and S∗|U1 = G.
2. If x ∈ Z for a stratum Z, then H i(S∗x) = 0 for i > p(Z).
3. For x ∈ Z ⊂ Xk, the attachment map αk : S∗k+1 → Rik∗S∗k is a quasi-isomorphism at x
up to p(Z).
Proposition 3.7. For a general perversity p, I pS∗(G0) satisfies the axioms (AX1)p,X,G⊗O.
Proof. Most of these properties follow, essentially, from the work in [22], especially the cone
formula, Proposition 2.1.
I pS i(G0) = 0 for i > n, since these sheaves would be defined by singular chains in I pCn−i,
which is clearly trivial for i > n.
The condition I pS i = 0 for i < 0 in property (1) is meant in the sense of the derived
category so that only the cohomology has to vanish in this range. That this happens can
be seen by induction on depth: If the depth of X is 0, X is a manifold, and for any x ∈ X,
H∗(I pS∗x;G0) ∼= H∞n−∗(Rn;G), which is certainly 0 for ∗ < 0. Now, assuming the condition of
depth d− 1, if X has depth d and x ∈ Xn−k, then H i(I pS∗(G0)x) ∼= I pH∞n−i(Rn−k × cL;G0),
which, depending on the perversity and i, is either 0 or I pHk−1−i(L;G0). By the induction
hypothesis, the intersection sheaf I pS∗L(G0) on L is quasi-isomorphic to 0 in negative dimen-
sions, which also implies by the hypercohomology spectral sequence that I pHk−1−i(L;G0) ∼=Hi(L; I pS∗L) is 0 for i < 0. Thus H∗(I pS∗(G0)x) = 0 for ∗ < 0, and we are finished by
induction.
Using excision, the restriction to U1 is quasi-isomorphic to the sheaf complex of ordinary
singular chains with coefficients in G, which in turn is quasi-isomorphic to G ⊗ O since U1
is an n-manifold (note that with our indexing conventions, degree 0 of a sheaf complex
corresponds to degree n of the singular chain complex).
Finally, for x ∈ Z ⊂ Xn−k, Hi(I pS∗x) ∼= lim−→x∈U I
pH∞n−i(U ;G0) ∼= I pH∞n−i(Rn−k × cL;G0).
Properties 2 and 3 now hold by applying the Kunneth formula for products with Rn−k and
Proposition 2.1 (the relation between the cone formula and the attaching map is explained
more fully in [22]).
The second proposition makes use of our new generalized truncation.
Proposition 3.8. For a general perversity p, any sheaf complex satisfying the axioms
(AX1)p,X,G is quasi-isomorphic to Q∗p(G).
Proof. Substituting the appropriate τF≤P for τ≤m, the proof is analogous to that in the usual
case for GM perversities; see, e.g., [7, Theorem 2.5]. We run through the argument for
completeness. Let S∗ be any sheaf complex satisfying the axioms. We let S∗k denote S∗|Uk.
We proceed by induction by showing that S∗k+1∼= τ
Xn−k
≤p Rik∗S∗k .
On U1, by property (1), we know that S∗|U1∼= G ∼= Q∗|U1 . This last quasi-isomorphism
is easy to check by repeated application of the third property of Lemma 3.3 and the fact
that i∗Ri∗A∗ and A∗ are always quasi-isomorphic when i is an inclusion of an open subset.
Using Lemma 3.3, we form the diagram
17
τXn−k
≤p S∗k+1
τXn−k
≤p αk- τ
Xn−k
≤p Rik∗S∗k
S∗k+1
? αk - Rik∗S∗k .?
The lefthand vertical map is the identity over Uk and a quasi-isomorphism over Xn−k, using
the third property of Lemma 3.3 and the assumption that S∗ satisfies condition (2) of the
axioms. Furthermore, using property (3) of the axioms, τXn−k
≤p αk is also a quasi-isomorphism:
on Uk, it restricts to the identity (in the derived category), and on the stratum Z ⊂ Xn−k it
restricts to τ≤p(Z)(S∗k+1|Z)→ τ≤p(Z)((Rik∗S∗k)|Z), which again is a quasi-isomorphism because
αk is a quasi-isomorphism over Z for ∗ ≤ p(Z) by assumption. Thus, utilizing the left side
and the top of the diagram, we obtain a quasi-isomorphism between S∗k+1 and τXn−k
≤p Rik∗S∗k .
The proposition follows by induction.
Together, the propositions prove Theorem 3.6.
Proposition 3.8 and Theorem 3.6 together yield the following corollary, which is a version
of Lemma 2.4 in the sheaf world. This corollary says that any generalized Deligne sheaf is
equivalent to an efficient-perversity Deligne sheaf.
Corollary 3.9. For any general perversity p and coefficient system G on X −Xn−1, Q∗p(G)
is quasi-isomorphic to Q∗p(G), where p is the efficient perversity associated to p.
3.1 Comparison with intermediate extensions
The machinery of perverse sheaves developed axiomatically by Beilinson, Bernstein, and
Deligne in [5] also contains a method for creating sheaf complexes that satisfy the intersection
homology axioms AX1. For background on perverse sheaves, we refer the reader to [5], [37,
Chapter X], [2, Chapter 7], [6], and [18, Chapter 5].
Let U ⊂ X be an open subset of X that is a union of strata, let i : U ↪→ X be the
inclusion, and let S∗ be a P -perverse sheaf on U for some general perversity P . Then there
is defined axiomatically in [5] the “intermediate extension functor” i!∗ such that i!∗S∗ is
the unique extension in the category of P -perverse sheaves of S∗ to X (meaning that the
restriction of i!∗S∗ to U is quasi-isomorphic to S∗) such that for each stratum Z ⊂ X − Uand inclusion j : Z ↪→ X,we have Hk(j∗i!∗S∗) = 0 for k ≥ P (Z) and Hk(j!i!∗S∗) = 0 for
k ≤ P (Z). We refer the reader to [5, Section 1.4] or [18, Section 5.2] for the precise definition
of the functor i!∗.
In particular, suppose we let U = X − Xn−1, that S∗ is just the local system G, and
that p is a general perversity on X. The sheaf G is certainly P -perverse on U with respect
to the perversity P (U) = 0. Now let P (Z) = p(Z) + 1. It follows that for each singular
stratum inclusion j : Z ↪→ X, we have Hk(j∗i!∗G) = 0 for k > p(Z) and Hk(j!i!∗G) = 0
18
for k ≤ p(Z) + 1. In the presence of the first condition, the second condition is equivalent
to the attachment map at the stratum Z being an isomorphism up through degree p(Z);
see [7, page 87]. But according to the axioms AX1, these conditions are satisfied by the
perversity p Deligne sheaf, which is also easily seen to be P -perverse. Thus, since i!∗G is the
unique extension of G with these properties, i!∗G is none other than the Deligne sheaf (up to
quasi-isomorphism). Thus we can also think of the generalized Deligne process as a means
to provide a concrete realization of i!∗G (which is defined quite abstractly and axiomatically
in [5]).
4 Constructibility and duality
Having established the equivalence in the realm of perverse sheaves between generalized
Deligne sheaves and intermediate extensions of coefficient systems, constructibility and dual-
ity results follow (at least with field coefficients) from the abstract perverse sheaf machinery
of [5]. However, we will provide (at least a sketch of) more direct proofs in the spirit of
Goresky-MacPherson [29] and Borel [7].
4.1 Constructibility
We refer to [7, Section V.3] for the definitions of X-clc (X-cohomologically locally constant),
X-cc (X-cohomologically constructible), and cc (cohomologically constructible), where X
refers to the stratification of X. These concepts are used briefly in the proof of Theorem
4.3, below, so the following proposition is worth having, but overall these concepts play a
minor role here, so we choose not to go into too much detail. We assume that the coefficient
system G on X −Xn−1 is a system of finitely-generated R-modules for a Noetherian ring R
of finite cohomological dimension.
Proposition 4.1. Any sheaf satisfying AX1p,X,G with respect to the general perversity p is
constructible. More particularly, it is X-clc and X-cc with respect to the given stratification
X of X, and it is cc.
Sketch of proof. This follows by induction over the strata just as for the traditional case
(e.g. [7, Proposition 3.12]), which relies nowhere on the particular form of the perversity.
We refer the reader there for thorough details. We simply note that, just as in that proof,
it suffices for us, by Proposition 3.7 and 3.8, to consider only the Deligne sheaves Q∗, and
then it follows by general sheaf theory and the properties of the pseudomanifold X that the
functor Rik∗ preserves the property of being X-cc [7, Corollary V.3.11.iii]. Similarly, it is
not hard to see that being X-cc is preserved by our generalized truncation functors, using
Lemma 3.3.
4.2 Duality
In this section, we show that the generalized Deligne sheaves with complementary general
perversities are Verdier dual to each other. This implies Goresky-MacPherson duality on
19
intersection homology for complementary general perversities. In fact, we will consider the
more general duality with coefficients in a principal ideal domain considered by Goresky and
Siegel [30]. For this, we need a definition.
Definition 4.2. Let R be a PID, and let E be a local coefficient system on X − Xn−1 of
finitely-generated free modules over R. We say that the pseudomanifold X is locally (p,E)-
torsion free if for all singular strata Z and each x ∈ Z, I pHccodim(Z)−2−p(Z)(Lx; E0) is R-torsion
free, where Lx is the link of x in X.
This definition is a direct analogue of the definition of locally p-torsion free in Goresky-
Siegel [30]. Note that any X is automatically locally (p,E)-torsion free if R is a field.
Let Q∗p(E) be the generalized Deligne sheaf of perversity p with coefficients E . Let t be
the top perversity, t(Z) = codim(Z)− 2. Let DY denote the Verdier dualizing functor over
the space Y - see [7, 29, 2] for details. Recall also that A∗[n] denotes the shifted complex
such that (A∗[n])i = Ai+n.
Theorem 4.3. Let p and q be general perversities such that p + q = t, and let X be an n-
dimensional stratified pseudomanifold that is locally (p, E)-torsion free. Let U1 = X −Xn−1.
Then DXQ∗p(E)[−n] ∼= Q∗q(DU1E [−n]) in the derived category of sheaves on X.
This is the usual duality statement in intersection homology (see [7, Theorem V.9.8]).
The principal novelty is the generality of the perversities and the acceptability of codimension
one strata.
Proof. The standard proof for GM perversities, given in [29] and [7] proceeds by showing that
DXQ∗p(E)[−n] satisfies the axioms AX2q,X,DU1E[−n], which, when p and q are GM perversities,
are equivalent to AX1q,X,DU1E[−n]. However, when p and q are general perversities, the axioms
AX2 are no longer well-formulated. In particular, when p is a GM perversity, AX2 utilizes
the notion p−1(j) = min{c | p(c) ≥ j} and is formulated in terms of conditions such as
“dim supp(HjS∗) ≤ n − p−1(j) for all j > 0.” But when we allow p to take different
values on different strata of the same dimension, it is no longer clear that such an equivalent
formulation of these axioms is possible. Thus instead we will show that DXQ∗p(E)[−n]
satisfies the axioms AX1q,X,DU1E[−n], which suffice by Propositions 3.8. This will be similar
to the proof of our main theorem of [23], though simpler since the context there called for
much more general spaces.
Let E = E ⊗ O, where O is the orientation sheaf on X − Xn−1. Using Lemma 2.4 and
Corollary 3.9 and noting that if p + q = t, then p + q = t, where p, q are the efficient
perversities associated to p, q, it suffices to prove the theorem when p and q are efficient. By
Propositions 3.7 and 3.8, we are free to interpret Q∗p(E) as either the generalized Deligne
sheaf or the sheaf of intersection chains I pS∗(E0), since we have seen that these sheaves are
quasi-isomorphic, making them equivalent in the derived category in which we work. It will
be useful to be able to use both points of view. Since Q∗p(E) is X-cc and cc by Proposition 4.1,
DXQ∗p(E) is also X-cc and cc by [7, Corollary V.8.7 and Proposition V.3.10.e]. In particular,
DXQ∗p(E) is X-clc, which is part of the definition of X-cc (see [7, Section V.3]).
20
As we work in the derive category, we are free to replace the condition S i = 0 for i < 0
in axiom (1) with the condition H∗(S∗) = 0 for i < 0 - see [7, Remark V.2.7.b].
The coefficient part of axiom (1) follows by the usual arguments: Let U1 = X − Xn−1,
and let i : U1 ↪→ X be the inclusion. Since this is an inclusion of an open set i∗ = i! [2, p. 62],
and, so i∗DX = i!DX = DU1i∗ [2, Proposition 3.4.5]. Thus the restriction of DXQ∗p(E)[−n]
to U1 is (DU1i∗Q∗p(E))[−n], which is quasi-isomorphic to (DU1E)[−n], using axiom (1) for
Q∗p(E).
Next, let x ∈ Z ⊂ Xn−k − Xn−k−1 and consider H∗(D∗X(I pS∗(E0))x[−n]), which is
isomorphic to lim−→x∈U H∗−n(U ;D∗x(I pS∗(E0))). For any sheaf complexA∗ over R in the derived
category Db(X) and any open set U ∈ X, we have an exact sequence
0→ Ext(Hi+1c (U ;A∗), R)→ H−i(U ;DXA∗)→ Hom(Hi
c(U ;A∗), R)→ 0 (1)
(see [2, Section 3.4]). Thus taking A∗ = I pS∗(E0) and shifting indices, there is an exact
sequence
0→ Ext(IHc∗−1(U ; E0), R)→ H∗−n(U ;D∗(I pS∗(E0)))→ Hom(IHc
∗(U ; E0), R)→ 0.
Since the distinguished neighborhoods of any point in a pseudomanifold constitute a cofinal
system of neighborhoods, we may assume that U ∼= Rn−k× cLk−1. Now, we may employ the
the Kunneth formula with Rn−k and the cone formula of Proposition 2.1 to conclude that
IHc∗(U ; E) = 0 for ∗ > k−2− p(Z) = k−2− (k−2− q(Z)) = q(Z). Furthermore, since X is
locally (p,E)-torsion free, Ext(IHck−2−p(Z)(U ; E), R) is also 0, so H∗−n(U ;D∗(I pS∗(E0))) = 0
for ∗ > q(Z). It is also clear that these groups must be 0 for ∗ < 0. This completes the
demonstration of axioms (1) and (2).
Finally, we must verify the attaching axiom, axiom (3). If x ∈ Z, this axiom is equivalent
to the condition that H i(S∗)x ∼= lim−→x∈U Hi(U −U ∩Z;S∗) for all i ≤ p(Z) (see [7, V.1.7], [2,
Section 4.1.4]), where U runs over all open neighborhoods of x. Again, we can limit ourselves
to a cofinal system of distinguished neighborhoods, and it suffices to find then isomorphisms
Hj(U ;D∗(I pS∗(E0))[−n]) → Hj(U − U ∩ Z;D∗(I pS∗(E0))[−n]) that are functorial in that
they commute with further restrictions U → V . By [23, Appendix], there is a map of short
exact sequences
21
0 0
Ext(Hn−j+1c (U ; I pS(E0)), R)
?- Ext(Hn−j+1
c (U − U ∩ Z; I pS∗(E0)), R)?
Hj(U ;D∗X(I pS∗(E0))[−n])?
- Hj(U − U ∩ Z;D∗X(I pS∗(E0))[−n])?
Hom(Hn−jc (U ; I pS(E0)), R)
?- Hom(Hn−j
c (U − U ∩ Z; I pS∗(E0)), R)?
0?
0?
,
where the maps of the top and bottom terms are induced by the inclusion maps I pHcj (U −
U ∩Xn−k; E0)→ I pHcj (U ; E0). Again since U ∼= Rn−k× cL, we can use the Kunneth theorem
with Rn−k to obtain that Hn−∗c (U − U ∩ Z; I pS(E0)) ∼= I pHc
∗(U − U ∩ Z; E0) is isomorphic
to I pHc∗(L; E0) and Hn−∗
c (U ; I pS(E0)) ∼= I pHc∗(U ; E0) ∼= I pHc
∗(cL; E0). By the cone formula,
the inclusion I pHc∗(L; E0) → I pHc
∗(cL; E0) is an isomorphism for ∗ < k − 1 − p(Z). Thus,
by the five lemma, Hj(U ;DX(I pS(E0))[−n]) → Hj(U − U ∩ Z;DX(I pS∗(E0))[−n]) is an
isomorphism for j ≤ q(Z). Since this computation is functorial with respect to restrictions,
we obtain the desired isomorphism in the limits.
Thus D∗(I pS∗(E))[−n] satisfies the axioms AX1q,X(D∗U1(E)[−n]), which completes the
proof of the theorem.
Corollary 4.4. Let X be a locally (p,E)-torsion free n-dimensional stratified pseudomanifold,
possibly with codimension one strata, where E is a local coefficient system on X −Xn−1 of
finitely-generated free modules over the principal ideal domain R. Let O be the orientation
sheaf of the n-manifold X −Xn−1. Let p and q be dual efficient perversities (p(Z) + q(Z) =
codim(Z)−2). Let TH∗ and FH∗ denote, respectively, the R-torsion subgroup and R-torsion
free quotient group of IH∗, and let Q(R) denote the field of fractions of R. Then,
1. Suppose that Hom(T pHci−1(X; E0), Q(R)/R) is a torsion R-module (in particular, if
T pHci−1(X; E0) is finitely generated). Then
Hom(F pHci (X; E0), R) ∼= F qH∞n−i(X; Hom(E , RX−Xn−1)0 ⊗O0)
22
and
Hom(T pHci−1(X; E0), Q(R)/R) ∼= T qH∞n−i(X; Hom(E , RX−Xn−1)0 ⊗O0).
2. If E is a local coefficient system on X−Xn−1 of finitely-generated F-modules for a field
F, then
Hom(I pHcn−i(X; E0); F) ∼= I qH∞i (X; Hom(E ,FX−Xn−1)0 ⊗O0).
3. When X is compact and orientable, we obtain as a special case the simpler, but more
familiar, special case
Hom(I pHi(X; Q0),Q) ∼= I qHn−i(X; Q0).
If, in addition, X is locally (p,Z)-torsion free, we have
Hom(F pHi(X; Z0),Z) ∼= F qHn−i(X; Z0) and Hom(T pHi(X; Z0),Q/Z) ∼= T qHn−i(X; Z0).
Proof. These statements follow direction from the theorem, using the universal coefficient
short exact sequence (1). See [7, 30, 23] for more details.
Intersection Lefschetz duality. These duality theorems imply easy proofs of Lefschetz-
type duality theorems on the intersection homology of pseudomanifolds with boundary. The
proof is reminiscent of the standard proof that intersection homology duality implies classical
Lefschetz duality on manifolds.
Let X be a compact oriented n-pseudomanifold with collared boundary ∂X (we leave
the non-compact and/or non-orientable cases to the reader). In other words, X − ∂X is a
pseudomanifold as defined above, ∂X is an n−1 pseudomanifold, and there is a neighborhood
of ∂X in X stratified homeomorphic to ∂X × [0, 1), where [0, 1) is unstratified. We do
not consider ∂X to be a codimension one stratum of X, but rather we let it inherit its
stratification from that ofX. This is the traditional convention for discussing pseudomanifold
bordism; see e.g. Siegel [49]. Then our duality theorems can be interpreted to give a duality
between I pH∗(X) and I qH∗(X, ∂X) (where “duality” is interpreted as providing a set of
results akin to those in Corollary 4.4, i.e. a set of intersection pairings and linking pairings
between the relevant torsion subgroups and torsion-free quotients across the appropriate
dimensions, provided X is appropriately locally torsion-free).
To see this, let X = X ∪∂X c∂X, the space obtained by adjoining to X a cone on the
boundary (or, equivalently, pinching the boundary to a point). Let v denote the vertex
of the cone point. Let p, q be dual general perversities on X, and let p−, q+ be the dual
perversities on X such that p−(Z) = p(Z) and q+(Z) = q(Z) for each stratum Z of X,
p−({v}) = −2, and q+({v}) = n. Our duality theorems provide a duality, in the above
sense, between I p−H∗(X) and I q+H∗(X). But now we simply observe that I p−H∗(X) ∼=I p−H∗(X − {v}) ∼= I pH∗(X), because the perversity condition at v ensures that no singular
simplex of a relevant dimension may intersect v. On the other hand, since I q+H∗(c∂X) = 0
by Proposition 2.1, I q+H∗(X) ∼= I q+H∗(X, c∂X) by the long exact sequence of the pair, but
I q+H∗(X, c∂X) ∼= I q+H∗(X, ∂X) ∼= I qH∗(X, ∂X) by excision.
23
Classical Lefschetz duality results via codimension one strata. Another interesting
feature of our extended duality theorems is that they provide a proof of the Lefschetz duality
theorem for manifolds with boundary M without needing to employ the added vertex trick
of the preceding paragraphs. This is not a direct feature of traditional intersection homology
theory. More generally, we obtain duality results of Lefschetz type for pseudomanifolds with
codimension one strata; these are not available in traditional intersection homology.
Let us see how this works for a manifold with boundary. Let M be a compact oriented n-
manifold with collared boundary, and let (M,∂M) provide the stratification. For simplicity,
assume that ∂M is connected, though we will loosen this condition in a moment. Then there
are only two efficient perversity possibilities for ∂M , say p(∂M) = 0 and q(∂M) = −1, and
these perversities are dual because −1 = p(∂M) + q(∂M) = t(∂M) = codim(∂M)− 2 = −1.
Now we claim that the dual intersection homology groups I qH∗(M ;R0) and I pH∗(M ;R0)
are respectively isomorphic to H∗(M ;R) and H∗(M,∂M ;R).
First, we note that, by the proof of Lemma 2.4, I qC∗(M ;R0) is quasi-isomorphic to
I q′C∗(M ;R0), where q′(∂M) = m for any negative integer m. It follows then easily from the
definition of allowability that I qC∗(M ;R0) is quasi-isomorphic to Cc∗(M − ∂M ;R), which is
quasi-isomorphic to C∗(M ;R) by homotopy equivalence. Secondly, it also follows Lemma
2.4 that I pC∗(M ;R0) is quasi-isomorphic to I p′C∗(M ;R0), where p′(∂M) is any non-negative
integer. In particular, taking p′(∂M) = 1, all chains become allowable, and we see that
I p′C∗(M ;R0) ∼= C∗(M ;R0) ∼= C∗(M,∂M ;R). This last isomorphism is discussed further in
Section 5.1.
Hence the duality of intersection homology becomes the dual pairing between H∗(M ;R)
and H∗(M,∂M ;R).
It is not hard to generalize this reasoning to see that we can also obtain the Lefschetz
pairing in its stronger form: Suppose M is a compact oriented n-manifold with boundary
M = M1∪M2, where each Mi is a union of connected components of the boundary. Suppose
that p(Z) = 0 if Z is a connected component of M1 and p(Z) = −1 if Z is a connected
component of M2. Let q = t− p. Then the duality between I pHi(M ;R0) and I qHj(M ;R0)
translates into the duality between H∗(M,M1;R) and H∗(M ;M2;R).
Remark 4.5. An anonymous referee has points pointed out an alternative proof of Lefschetz
duality for intersection homology. The author is unaware of this proof in the literature, so
it is reproduced here for the interested reader:8
Let (Xn, ∂X) be a compact oriented pseudomanifold with boundary, i : X−∂X ⊂X, j : ∂X ⊂ X. For S∗ a sheaf complex on X, the distinguished triangle
i!i∗S∗ → S∗ → Rj∗j
∗S∗ +1−→
induces on hypercohomology the exact sequence of the pair (X, ∂X) with coeffi-
cients in S∗:
· · ·Hk(X, ∂X;S∗)→ Hk(X;S∗)→ Hk(∂X;S∗)→ Hk+1(X, ∂X;S∗) · · · .8We have made some changes to notation for consistency with the present paper, but otherwise we quote
directly.
24
Let I pC∗ and I qC∗ be complementary [perversity] intersection chain sheaves on
the interior of X − ∂X. Setting S∗p = Ri∗I pC∗ and S∗q = Ri∗I qC∗, we get
Hk(X, ∂X;S∗p ) = Hk(i!i∗S∗p )
= Hk(i!I pC∗)= Hk(i!DI
qC∗[−n])
= Hk−n(D(i∗I qC∗))= Hom(Hn−k(X;S∗q ),Q),
which is the desired Poincare-Lefschetz result.
4.3 Sheaf pairings
In this section, we present the following theorems.
Theorem 4.6. Given a pairing of local systems m1 : E ⊗ F → G on X −Xn−1 and general
perversities such that p(Z) + q(Z) ≤ r(Z) for all singular strata Z, then in the bounded
derived category Db(X), there is a unique morphism m : Q∗p(E∗)L⊗ Q∗q(F) → Q∗r(G) that
restricts to m1 on X − Xn−1. Furthermore, if r ≤ t and G = O, the orientation sheaf of
X −Xn−1, then there exists a pairing m : Q∗p(E∗)L⊗Q∗q(F) → D∗X [−n] that restricts to m1
on X −Xn−1, where D∗X [−n] is the shift of the Verdier dualizing complex on X.
Theorem 4.7. If q = t−p, then the pairing on intersection homology induced by the pairings
of Theorem 4.6 coincide with those of Corollary 4.4.
Remark 4.8. Note that we do not require the perversities in these theorems to be efficient,
and even if p and q are efficient, p+ q may not be. Of course in the statements we can always
replace, for example, Q∗r by Q∗r, using Corollary 3.9.
We will explain in Remark 4.10 why we must have r ≤ t in the second statement of
Theorem 4.6.
Once again, the details of the proofs are mostly those that can be found already in
Chapter V.9.C of Borel’s book [7]. In this case, we will leave it to the reader to verify
that Borel’s treatment holds in our context. We simply note that the crucial point of
generalization is the following modification to [7, Lemma V.9.1].
Lemma 4.9. Let Y be a topological space and A∗ ∈ D(Y ), the derived category of sheaves
on Y . Let F be a locally-finite collection of subsets of Y , and let P be a function F → Z.
Suppose that the natural inclusion τF≤PA∗ ↪→ A∗ is a quasi-isomorphism. Then,
1. for any B∗ ∈ D(Y ), the natural homomorphism
MorD(Y )(A∗, τF≤PB
∗)→ MorD(Y )(A∗,B∗)
is an isomorphism, and
25
2. if U = Y − |F| is an open subset of Y and i : U ↪→ Y is the inclusion, then for any
B∗ ∈ D(Y ),
MorD(Y )(A∗, τF≤PRi∗B
∗)→ MorD(Y )(i∗A∗,B∗)
is an isomorphism.
Proof. The proof is the same as that of [7, Lemma V.9.1], utilizing the properties of τF≤P in
place of those of the standard truncation.
As in [7], one used the lemma to extend the relevant morphisms by induction and uniquely
from the coefficient pairings on X −Xn−1 up to the full pairings on X.
Remark 4.10. In order for these extension arguments to work, the hypotheses of Lemma 4.9
must be satisfied in the appropriate contexts. The reason that we must have r ≤ t in the
second statement of Theorem 4.6 is that we cannot apply Lemma 4.9 when p + q > t and
the desired codomain is D∗X [−n].
In slightly more detail, the idea of the proof of Theorem 4.6 is to show that on a given
Uk+1 = X −Xn−k−1, (Q∗pL⊗Q∗q)|Uk+1
will be quasi-isomorphic to τXn−k
≤r (Q∗pL⊗Q∗q)Uk+1
for any
r ≥ p + q. Of course we also have Q∗r|Uk+1= τ
Xn−k
≤r Rik∗Q∗r|Ukby the definition of Q∗. So
an extension argument can be made using the lemma. However, by [7, Lemma V.9.3], on
Uk+1, D∗X [−n]|Uk+1is only quasi-isomorphic to Rik∗D
∗X [−n]Uk
up to degree k− 2 = t(Xn−k).
Hence if p+ q > t, the extension argument will not work.
At the level of chains, the issue is that for any p ≤ t, we know from [22, Proposition
2.1] that I pC∗(X;R0) is isomorphic to I pC∗(X;R), which is a subgroup of C∗(X;R), whose
homology groups, up to the proper re-indexing, are the cohomology groups of DX [−n],
assuming X is oriented. So, there is a well-defined map I pC∗(X;R0)→ C∗(X;R). However,
if p(Z) > t(Z) for any singular stratum Z, then we will not necessarily have an isomorphism
from I pC∗(X;R0) to I pC∗(X;R), and hence no obvious chain map I pC∗(X;R0)→ C∗(X;R),
only a map I pC∗(X;R0)→ C∗(X;R0).
5 Intersection pairings on piecewise linear pseudoman-
ifolds
In this final section, we look at the intersection pairing of chains on a piecewise linear (PL)
pseudomanifold and see how the classical Goresky-MacPherson pairing of [28] and the more
general pairings of [25] extend to general perversities.
Throughout this section, X will be a compact PL pseudomanifold, possibly with codi-
mension one strata. In particular, this means that X possesses a family of triangulations
compatible with the stratification in the sense that each Xk will be a subcomplex of any
triangulation. More details on PL pseudomanifolds can be found in [7, Chapters I,II]. In
this section, we will consider only the PL chain complexes.
26
5.1 Some basics concerning PL chains and stratified coefficients
Recall that the PL chain complex Cc∗(X) is defined to be lim−→Cc,T
∗ (X), where Cc,T∗ (X) is
the simplicial chain complex based on the triangulation T and the limit is taken over all
compatible triangulations of X. In particular, every PL chain can be described in a specific
triangulation. This definition generalizes in the obvious way to closed supports to yield
C∞∗ (X).
We can also extend the PL chain complex to include stratified coefficient systems G0. In
fact, we observe that any simplex σ of a triangulation is either completely contained in Xn−1,
in which case it inherits an automatic 0 coefficient, or it intersects Xn−1 only in its boundary,
in which case a unique coefficient lift can be assigned to σ by determining what happens
at one interior point. Furthermore, the boundary of the simplex with its coefficient is well-
defined - any boundary faces in Xn−1 vanish due to the 0 coefficient strata and the other faces
inherit their coefficients from σ. Thus if X is a PL pseudomanifold, it is possible to define
the simplicial chain complex C∗(X;G0), as well as the subcomplexes I pC∗(X;G0) for any
general perversity. These are defined precisely as above in Section 2, though the allowability
condition for a PL i-simplex σ can be simplified to dim(σ ∩ Z) ≤ i− codim(Z) + p(Z).
We form the subcomplexes and quotient complexes C∗(U ;G0) and C∗(X,U ;G0) in the
obvious way. We generally continue to use G0 also to denote the restriction of this coefficient
system to subsets. If ξ is a chain in Ci(X;G0) represented by a chain in CTi (X;G0) for some
triangulation T , we let the support |ξ| be the union of all i-simplices σ of T such that the
coefficient of σ in ξ is not identically 0.
Notice that in the PL setting we obtain for any perversity the nice feature we observed
previously in Section 2.3 for efficient perversities: the coefficient of any i-simplex σ is de-
fined by the lift of a single point σ−1(X − Xn−1), which includes the entire interior of ∆i.
A useful consequence of this is that if we use constant coefficients on X − Xn−1, then we
have C∗(X;G0) ∼= C∗(X,Xn−1;G), where the isomorphism assigns to any gσ ∈ C∗(X;G0)
such that g is not identically 0 the obvious extension of g to the constant system G on X;
conversely, any gσ + C∗(Xn−1;G) ∈ C∗(X,X
n−1;G) for which |σ| 6⊂ Xn−1 determines an
element of C∗(X;G0) by restricting the coefficient lift to σ−1(X − Xn−1). The homomor-
phisms thus determined are obviously mutual inverses and encompass all generators. Thus,
we may think of I pC∗(X;G0) ⊂ C∗(X;G0) as being a subcomplex of C∗(X,Xn−1;G) when
it is useful to do so.
Pursuing this idea slightly further, there is also a natural homomorphism, though it is not
a chain map, ρ : C∗(X;G0)→ C∗(X;G). We define ρ as the composition of the isomorphism
C∗(X;G0) ∼= C∗(X,Xn−1;G) with the standard splitting C∗(X,X
n−1;G) → C∗(X;G) that
takes gσ + C∗(Xn−1;G) ∈ C∗(X,X
n−1;G) such that |σ| 6⊂ Xn−1 to gσ ∈ C∗(X;G). This
assignment is clearly additive, yielding a homomorphism. We observe that |ξ| = |ρ(ξ)|.However, it is not necessarily true that |∂ρ(ξ)| = |ρ(∂ξ)|; for example, if the boundary of
ρ(ξ) is in Xn−1, then we might have ∅ 6= |∂ρ(ξ)| ⊂ Xn−1 but ρ(∂ξ) = ρ(0) = 0. Thus ρ will
not in general be a chain map.
Also, notice that |ρ(ξ)| ∩ (X −Xn−1) = |ξ| ∩ (X −Xn−1), and |∂ρ(ξ)| ∩ (X −Xn−1) =
|ρ(∂ξ)| ∩ (X −Xn−1) = |∂ξ| ∩ (X −Xn−1).
27
To simplify notation below, we will often denote ρ(ξ) by ξ.
The correspondence between chains and homology. Given closed PL subspaces A ⊃B such that dim(A) = i, dim(B) < i, it is well-known that there is a bijection between
Hi(A,B;G) and the set of chains ξ ∈ C∞∗ (X;G) such that |ξ| ⊂ A and |∂ξ| ⊂ B (see
[31, 28, 42]). We observe here a relative version of this phenomenon that includes stratified
local coefficients:
Lemma 5.1. Suppose that A ⊃ B are closed PL subspaces of X such that dim(A − A ∩Xn−1) ≤ i and dim(B−B∩Xn−1) ≤ i−1. Let A denote the closure of A−A∩Xn−1 in X (or,
equivalently, the union of all simplices σ in a triangulation of A such that σ∩(X−Xn−1) 6= ∅)and similarly for B. Then there is a bijection between Hi(A,B;G0) ∼= Hi(A, B;G0) and the
set of chains ξ ∈ Ci(X;G0) such that |ξ| ⊂ A and |∂ξ| ⊂ B. Furthermore, if G is a
constant coefficient system, then there is a further bijection to Hi(A∪Xn−1, B ∪Xn−1;G) ∼=Hi(A,B ∪ (A ∩Xn−1);G) ∼= Hi(A, B ∪ (A ∩Xn−1);G).
Proof. The isomorphism Hi(A,B;G0) ∼= Hi(A, B;G0) is immediate because simplices sup-
ported in Xn−1 must carry a 0 coefficient. The isomorphisms Hi(A∪Xn−1, B ∪Xn−1;G) ∼=Hi(A,B ∪ (A ∩Xn−1);G) ∼= Hi(A, B ∪ (A ∩Xn−1);G) are by excision. Furthermore, for a
constant coefficient system G, there is an evident isomorphism C∗(A∪Xn−1, B∪Xn−1;G) ∼=C∗(A,B;G0).
Now we slightly generalize an argument of McClure [42, Lemma 4.1]: Hi(A, B;G0) is
the ith homology group of C∗ = C∗(A;G0)/C∗(B;G0), which is the cycles of Ci modulo the
boundaries of Ci. But the group of cycles of Ci = Ci(A;G0)/Ci(B;G0) consists precisely of
those chains of C∗(X;G0) that are supported in A and whose boundaries are contained in
B. Note that any piece of the boundary that would be contained in Xn−1 vanishes due to
the coefficient system. But the subgroup of boundaries in Ci is ∂Ci+1(A;G0) + Ci(B;G0),
which is zero due to the dimension hypotheses and again the fact that all chains supported
in Xn−1 vanish.
Remark 5.2. In the situation of the preceding lemma and assuming constant coefficients G,
note that if ξ is a chain in Ci(X;G0) corresponding to a certain element of Hi(A, B ∪ (A ∩Xn−1);G), then the element of Ci(X;G) that traditionally corresponds to this homology
element is precisely ρ(ξ) = ξ.
5.2 The Goresky-MacPherson pairing
Now, we generalize to general perversities the Goresky-MacPherson intersection pairing of
[28]. For simplicity, we start with constant coefficients and an oriented pseudomanifold. Let
Xn be a compact oriented piecewise-linear pseudomanifold, possibly with codimension one
strata. C∗(X) and I pC∗(X) denote complexes of PL chains over a fixed coefficient ring R.
Let us first recall the intersection pairing for GM perversities p, q such that p+ q ≤ r ≤ t.
Suppose x ∈ I pCi(X;R) and y ∈ I qCj(X;R) are two chains such that the pairs (|x|, |y|),
28
(|∂x|, |y|), and (|x|, |∂y|) are each pairs of chains in stratified general position. Recall that |x|is the support of x, and that two subspaces A and B of X are in stratified general position
if A∩Z and B ∩Z are in general position in Z for each stratum Z, regular or singular. We
take the intersection product of x and y, following the procedure in [28] and generalized in
[25].
The chains x and y can be represented by homology classesHi(|x|, |∂x|;R) andHj(|y|, |∂y|;R)
(see [28, 7]). Let J = |∂x|∪|∂y|∪Xn−1. Then the Goresky-MacPherson intersection product
x t y is defined by first applying the following composition (all groups have R coefficients):
Hi(|x|, |∂x|)×Hj(|y|, |∂y|)
Hi(|x|, |x| ∩ J)×Hj(|y|, |y| ∩ J)?
Hi(|x| ∪ J, J)×Hj(|y| ∪ J, J)?
Hn−i(X − J,X − (|x| ∪ J))×Hn−j(X − J,X − (|y| ∪ J))
(−1)n(n−i)(· ∩ [X])−1 × (−1)n(n−j)(· ∩ [X])−1
?
H2n−i−j(X − J,X − ((|x| ∩ |y|) ∪ J))
∪
?
Hi+j−n((|x| ∩ |y|) ∪ J, J)
(−1)n(2n−i−j)(· ∩ [X])
?
Hi+j−n(|x| ∩ |y|, |x| ∩ |y| ∩ J).
∼=
?
(2)
Here the last isomorphism is by excision, [X] is the fundamental orientation class of X,
and ∩[X] is the cap product inducing the Whitehead-Dold-Goresky-MacPherson duality
isomorphism (see [28, Appendix]). The signs here were not present in the original Goresky-
MacPherson paper [28], but an argument is made in [25] that these signs are useful in order
for the duality to be a chain map in the appropriately graded sense.
29
The last step in the Goresky-MacPherson intersection product is to use the long exact
sequence of the triple, excision, and the fact that dim(|x|∩ |y|∩Xn−1) ≤ i+ j−n−2, which
follows from the perversity conditions and the stratified general position, to conclude that
Hi+j−n(|x| ∩ |y|, |x| ∩ |y| ∩ |J |) ∼= Hi+j−n(|x| ∩ |y|, (|∂x| ∩ |y|)∪ (|x| ∩ |∂y|)), which represents
a chain, the intersection product x t y, in Ci+j−n(X;R). In particular, the third term in the
exact sequence of (|x| ∩ |y|, |x| ∩ |y| ∩J, (|∂x| ∩ |y|)∪ (|x| ∩ |∂y|)) is isomorphic by excision to
H∗(|x|∩|y|∩Xn−1, ((|∂x|∩|y|)∪(|x|∩|∂y|))∩Xn−1), but if dim(|x|∩|y|∩Xn−1) ≤ i+j−n−2,
this term vanishes for ∗ > i+ j − n− 2, which is enough for the desired isomorphism.
The reason that dim(|x|∩ |y|∩Xn−1) ≤ i+ j−n−2 is that if p+ q ≤ r ≤ t, then for each
stratum Z, dim(|x| ∩Z) ≤ i− codim(Z) + p(Z) and dim(|y| ∩Z) ≤ j − codim(Z) + q(Z) so
that, if they are in stratified general position,
dim(|x| ∩ |y| ∩ Z) ≤ i− codim(Z) + p(Z) + j − codim(Z) + q(Z)− dim(Z)
= i+ j + p(Z) + q(Z)− codim(Z)− (codim(Z) + dim(Z))
≤ i+ j + t(Z)− codim(Z)− n= i+ j + codim(Z)− 2− codim(Z)− n= i+ j − 2− n.
Furthermore, by the same type of computations, we see that the simplices of x t y must
be r-allowable. Similarly, if (|∂x|, |y|) and (|x|, |∂y|) are in stratified general position, then
since ∂|x t y| ⊂ (|∂x| ∩ |y|) ∪ (|x| ∩ |∂y|), an analogous argument shows that ∂(x t y) is
allowable. Thus x t y ∈ I rC∗(X;R).
Now, suppose that p and q are general perversities, that x ∈ I pCi(X;R0) and y ∈I pCj(X;R0). In order to apply the pairings, we first translate x, y to elements of C∗(X;R)
by considering ρ(x) = x and ρ(y) = y as defined above. We can then proceed as before
to translate x, y into elements of Hi(|x|, |∂x|;R) and Hj(|y|, |∂y|;R) and then apply the
sequence of maps in diagram (2). The trouble is the last step, since in the general perversity
setting, it may no longer be true that dim(|x| ∩ |y| ∩ Xn−1) ≤ i + j − n − 2. In addition,
∂x may contain simplices in Xn−1 that do not occur in ∂x due to the stratified coefficient
system, which may also a priori cause some trouble.
However, letting again J = |∂x| ∪ |∂y| ∪Xn−1, we do still obtain a well-defined homology
class in Hi+j−n(|x| ∩ |y|, |x| ∩ |y| ∩J) ∼= Hi+j−n(|x| ∩ |y|, (|∂x| ∩ |y|)∪ (|x| ∩ |∂y|)∪ (|x| ∩ |y| ∩Xn−1)). Now, assume that the pairs (|x|, |y|), (|∂x|, |y|), and (|x|, |∂y|) are all in stratified
general position. Then, applying computations such as those above,
dim((|x| ∩ |y|) ∩ (X −Xn−1))
= dim((|x| ∩ |y|) ∩ (X −Xn−1))
≤ i+ j − n
and
dim(((|∂x| ∩ |y|) ∪ (|x| ∩ |∂y|) ∪ (|x| ∩ |y| ∩Xn−1)) ∩ (X −Xn−1))
= dim(((|∂x| ∩ |y|) ∪ (|x| ∩ |∂y|)) ∩ (X −Xn−1))
= dim(((|∂x| ∩ |y|) ∪ (|x| ∩ |∂y|)) ∩ (X −Xn−1)) ≤ i+ j − n− 1.
30
So by Lemma 5.1, our homology class corresponds to a chain in Ci+j−n(X;R0) ∼= Ci+j−n(X,Xn−1;R)
with support in the closure of |x| ∩ |y| ∩ (X − Xn−1) = |x| ∩ |y| ∩ (X − Xn−1) and with
boundary supported in the closure of ((|∂x| ∩ |y|) ∪ (|x| ∩ |∂y|)) ∩ (X −Xn−1). This latter
closure is contained in (|∂x| ∩ |y|) ∪ (|x| ∩ |∂y|), where the boundary is taken in C∗(X;R0),
while the former is contained in |x| ∩ |y|.Now, computations just like those above, applied to the pairs |x| ∩ |y|, |∂x| ∩ |y|, and
|∂y| ∩ |x| allow us to conclude that this chain is in I rCi+j−n(X;R0).
In summary, we have proven the following theorem:
Theorem 5.3. Let X be a compact oriented PL stratified n-pseudomanifold. Let p, q be
general perversities. Let x ∈ I pC∗(X;R0), y ∈ I qC∗(X;R0) be such that the pairs (|x|, |y|),
(|∂x|, |y|), and (|x|, |∂y|) are in stratified general position. Then the Goresky-MacPherson
intersection pairing yields a well-defined chain x t y ∈ I rC∗(X;R0) for any r ≥ p + q.
Furthermore, if r ≤ t, then x t y ∈ I rC∗(X;R).
The last statement uses the fact that I rC∗(X;R) ∼= I rC∗(X;R0) when r ≤ t - see
Section 2.2. Furthermore, using this identification, we see that this pairing reduces to the
usual Goresky-MacPherson pairing if p, q are traditional with p+ q ≤ t.
Local Coefficients. Theorem 5.3 can be generalized in the local coefficient/possibly non-
orientable case to the following:
Theorem 5.4. Let X be a compact PL n-pseudomanifold, let p, q be general perversities, and
let E and F be local coefficient systems on X −Xn−1 over a principal ideal domain R with
a pairing E ⊗ F → G. Let O be the R orientation coefficient system on X −Xn−1. Let x ∈I pC∗(X; E0), y ∈ I qC∗(X;O ⊗ F0) be such that the pairs (|x|, |y|), (|∂x|, |y|), and (|x|, |∂y|)are in stratified general position. Then the Goresky-MacPherson intersection pairing extends
to yield a well-defined chain x t y ∈ I rC∗(X;G0) for any r ≥ p+ q.
The arguments are mostly the same as those above, however we will need the following
version of the Whitehead-Dold-Goresky-MacPherson duality isomorphism.
Lemma 5.5. Suppose Xn−1 ⊂ D ⊂ C are closed PL subspaces of the compact, not nec-
essarily oriented, pseudomanifold Xn. Let G be a local coefficient system of R modules on
X − Xn−1, where R is a principal ideal domain. Let O be the local orientation system
with R coefficients on X − Xn−1. Then there is an isomorphism H i(X − D,X − C;G) →Hn−i(C,D;O0⊗G0) that is induced by inclusions and cap product with the fundamental class
[XR] ∈ Hn(X;O0).
Note that since X −D ⊂ X −Xn−1, H i(X −D,X − C;G) is well-defined with no need
for stratified coefficients.
We will turn to the proof of the lemma in a moment. Assuming it for now, we show why
Theorem 5.4 holds.
Firstly, since we have local coefficient systems, we cannot represent x and y as elements
of the form H∗(A,B;R) but only as elements of the form H∗(A,B; Υ0), where A, B denote
31
the supports of the relevant chains, ∗ = i or j, and Υ is one of E or O ⊗F , as appropriate.
However, there are nonetheless obvious inclusion morphismsHi(A,B; Υ0)→ Hi(A∪J, J ; Υ0).
Note that in this case we don’t even need to utilize x or y at all, so this also provides an
alternative approach to the discussion above.
Now, we can apply 5.5 to each term Lemma in place of the Whitehead-Dold-Goresky-
MacPherson duality of [28] to obtain cochains of the form H∗(X−B,X−A;O⊗Υ). The cup
product, using the pairing E ⊗F → G, gives us an element of H2n−i−j(X−J,X− (|x|∩ |y|∪J);O⊗G), and then applying duality again gives an element of Hi+j−n((|x| ∩ |y|)∪J, J ;G0).
The excision isomorphism to Hi+j−n(|x| ∩ |y|, |x| ∩ |y| ∩ J ;G0) applies as usual, and applying
Lemma 5.1, this corresponds to an element of Ci+j−n(X;G0), which we can again verify to
be an element of I rCi+j−n(X;G0).
This yields Theorem 5.4.
Now we return to Lemma 5.5.
Sketch of proof. Assume X is triangulated as a complex K such that C, D, and each skeleton
are subcomplexes. If A is a subcomplex of K, let A denote the complement of the open first
derived neighborhood of A in the derived triangulation K ′. We first note that H i(X−D,X−C;G) ∼= H i(D, C;G) by homotopy equivalences.
Now we use the standard simplicial duality arguments as in, e.g. [41, Chapter 5], which
generalize in evident ways to the local coefficient case. In particular, we can think of
Ci(D, C;G) as being generated by the i-cochains cσ,g that evaluate to 0 except on a sin-
gle i-simplex σ of K ′ supported in D but not in C, which evaluates to an element g ∈ Gx,where x is a point in σ. Since all such simplices are in the interior of the manifold X−Xn−1,
if we let z be a chain representing the orientation class [XR], then after passing to the second
derived subdivision, cσ,g∩z is represented by the dual block e(σ) to σ, carrying the coefficient
in O ⊗ G0 inherited from the evaluation of cσ,g and from the coefficient carried by z in a
neighborhood of σ (by working within a contractible neighborhood of σ, we can see that this
coefficient is well-defined). So ∩z takes the cochains of C∗(D, C;G) to the dual blocks (with
coefficients) of the simplices of D not supported in C, and, as shown in [41], this is a chain
isomorphism to the dual block complex Cn−∗(¯C, ¯D;O⊗G0); we are free to use the stratified
coefficients G0 since we never get close enough to Xn−1 for the 0 system on it to matter. This
chain isomorphism induces an isomorphism H i(D, C;G)→ Hn−i(¯C, ¯D;O⊗G0). If we do not
wish to think about dual blocks near the singular set Xn−1, we can alternatively think of
the pair ( ¯C, ¯D) as being the appropriate dual pair in the manifold double D∪−D (with the
appropriately extended coefficient systems), but then the resulting relative homology group
is isomorphic by excisions to our Hn−i(¯C, ¯D;O ⊗ G0) in X.
Finally, we note that ¯D strongly deformation retracts to D, and similarly for ¯C. Just as
for ordinary homology, it follows that H∗(¯D;O ⊗ G0) ∼= H∗(D;O ⊗ G0) and similarly for C.
Thus by the five lemma, Hn−i(¯C, ¯D;O ⊗ G0) ∼= Hn−i(C,D;O ⊗ G0).
32
5.3 The pairing algebra
Having established the basic results on the intersection pairing in the previous section, most
of the results on the algebraic properties of intersection pairings developed in [25] go through
without much extra effort. We will state the main theorems.
First, we must recall some definitions from [25]. Let X(k) denote the product of k copies
of X, endowed with the product stratification. Let k = {1, . . . , k}. If R : k → k′ is any
map of sets, it induces a map R∗ : X(k′) → X(k) by R∗(x1, . . . , xk′) = (xR(1), . . . , xR(k)).
These are generalizations of the diagonal map x → (x, x), which is induced by the unique
map 2→ 1.
If A is a PL subset of X(k), we will say that A is in stratified general position with respect
to R∗ if for each stratum Z ⊂ Xd1 × · · · ×Xdkof X(k) such that di = d` if R(i) = R(`), we
have
dim((R∗)−1(A ∩ Z)) ≤ dim(A ∩ Z) +k′∑i=1
dR−1(i) −k∑i=1
di. (3)
Note that the condition on the dis implies that dj = d` for any j, ` ∈ R−1(i) so that this
sum is well-defined.
In other words, A is in stratified general position with respect to R∗ if for each stratum Z
of X(k), A∩Z is in general position with respect to the map of manifolds from the stratum
containing (R∗)−1(Z) to Z. A PL chain is said to be in stratified general position if its
support is, and we write CR∗∗ (X(k)) for the subcomplex of PL chains D of C∗(X(k)) such
that both D and ∂D are in stratified general position with respect to R∗. Similarly, we let
CR∗∗ (X(k);R0) be those chains D such that |D| and |∂D| are in stratified general position.
For a differential graded complex C∗, SmC∗ is the shifted complex with (SmC∗)i = Ci−m
and ∂SmC∗ = (−1)m∂C∗ . For a chain ξ, we define |Smξ| = |ξ|.The product ε is the multilinear extension of the product that takes σ1 ⊗ σ2, where the
σi are oriented simplices, to a chain with support |σ1| × |σ2| and with appropriate orienta-
tion. This is a direct generalization of the standard simplicial cross product construction
(see e.g. [44]); we refer the reader to [42, Section 7] for details. With dim(X) = n, εk :
S−nC∗(X)⊗ · · · ⊗ S−nC∗(X)→ S−knC∗(X(k)) is defined to be (−1)e2(n,...,n) times the com-
position S−knε ◦Θ, where Θ is the appropriately signed chain isomorphism Θ : S−nC∗(X)⊗· · · ⊗ S−nC∗(X) → S−nk(C∗(X(k))) (see [25, Remark 3.2]). Here e2(n, . . . , n) is the ele-
mentary symmetric polynomial of degree two on the k symbols n, . . . , n, so e2(n, . . . , n) =∑ki=1
∑j<i n
2. In other words, ε is the composite
S−nC∗(X)⊗ · · · ⊗ S−nC∗(X)Θ
- S−nk(C∗(X)⊗ · · · ⊗ C∗(Xk))
(−1)e2S−nkε- S−nkC∗(X(k)).
ε and ε are monomorphisms. Furthermore, ε is a degree 0 chain map.
Unfortunately, all of this shifting and application of signs is necessary for the appropriate
pairings to be degree 0 chain maps.
33
We note that this definition also determines a map ε : S−nC∗(X,Xn−1)⊗· · ·⊗S−nC∗(X,Xn−1)→
S−nkC∗(X(k), X(k)nk−1), or, equivalently ε : S−nC∗(X;R0)⊗· · ·⊗S−nC∗(X;R0)→ S−nkC∗(X(k);R0).
This is a well-defined monomorphism because |ε(σ1 ⊗ · · · ⊗ σk)| ⊂ (X(k))nk−1 if and only if
|σi| ⊂ Xn−1 for some at least one i.
Next, recall the following definition from [25], generalizing that of [42], of the complex of
chains in general position.
Definition 5.6. For k ≥ 2, let the domain Gk be the subcomplex of (S−nC∗(X;R))⊗k
consisting of elements D such that both ε(D) and ε(∂D) are in stratified general position
with respect to all generalized diagonal maps, i.e.
Gk =⋂k′<k
⋂R:k�k′
ε−1(S−nkCR∗
∗ (X(k))).
It is shown in [25] that the inclusion Gk ↪→ (S−nC∗(X;R))⊗k is a quasi-isomorphism for
all k ≥ 1.
Furthermore, there are the intersection chain versions. If P = (p1, . . . , pk) is a collection
of GM perversities and GPk = Gk ∩ (S−nIC p1
∗ (X)⊗ · · · ⊗ S−nIC pk∗ (X)), then the inclusion
GPk ↪→ S−nIC p1
∗ (X)⊗ · · · ⊗ S−nIC pk∗ (X) is a quasi-isomorphism.
Here we will need also a relative version.
Definition 5.7. For k ≥ 2, let the domain Gk,0 be the subcomplex of (S−nC∗(X,Xn−1;G))⊗k
consisting of elements D such that both ε(D) and ε(∂D) are in stratified general position
with respect to all generalized diagonal maps, i.e.
Gk,0 =⋂k′<k
⋂R:k�k′
ε−1(S−nkCR∗
∗ (X(k);R0)).
First, we should observe that Gk,0 is a well-defined chain complex. If x, y ∈ Gk,0, then
certainly −x ∈ Gk,0, as well at x+ y because the sum of chains in general position will also
be in general position. Furthermore, if D ∈ Gk,0, then it is built into the definition that ∂D
will also be in stratified general position, and of course ∂∂D = 0; thus ∂D ∈ Gk,0. So Gk,0
is a chain complex.
Similarly, we can define the intersection chain versions:
Definition 5.8. For k ≥ 2 and a sequence of general perversities P = (p1, . . . , pk), let the
domain GPk,0 be the subcomplex of S−nI p1C∗(X;R0) ⊗ · · · ⊗ S−nI pkC∗(X;R0) consisting of
elements D such that both ε(D) and ε(∂D) are in stratified general position with respect to
all generalized diagonal maps, i.e.
GPk,0 = Gk,0 ∩
(S−nIC p1
∗ (X)⊗ · · · ⊗ S−nIC pk∗ (X)
).
Then we have the following theorem.
Theorem 5.9. The inclusion Gk,0 ↪→ (S−nC∗(X;R0))⊗k is a quasi-isomorphism for all
k ≥ 1, as are the inclusions GPk,0 ↪→ S−nIC p1
∗ (X;R0)⊗ · · · ⊗ S−nIC pk∗ (X;R0).
34
Sketch of proof. Ultimately, the proof is more-or-less the same as that of Theorem 3.5 of
[25], which shows that Gk is quasi-isomorphic to (S−nC∗(X))⊗k. The general idea is to push
chains by homotopies until all desired general positions are achieved. In fact, since ∂D ⊂ ∂D,
it suffices to use the arguments of the proof of [25, Theorem 3.5], which constructs all relevant
homotopies and homologies for D and ∂D. These are absolute homologies, but they become
relative homologies when considered with coefficients in R0.
Umkehr maps. The motivation in [42] for creating the complexes Gk for the manifold M
is that Gk serves as the domain for a generalized intersection pairing. Using this domain, the
intersection pairing can be defined as a chain map, rather than as an ad hoc construction
on pairs of chains in suitable general position. Furthermore, this intersection pairing on Gk
is used to show that the intersection pairing extends to the algebraic structure of a Leinster
partial commutative differential graded algebra (DGA). Similarly, in [25], the GPk are shown
to be domains for the intersection pairing of intersection chains, and these pairings are shown
to induce partial restricted commutative DGAs.
To extend these results here, we need to consider the appropriate relative form of the
general intersection pairings.
The more general intersection homology multi-products come from considering umkehr
maps on pseudomanifolds. This was done in [25, Section 4.2]. Here we consider the relative,
or stratified coefficient, version, which in some sense is simpler, just as for the Goresky-
MacPherson pairing we were able not to concern ourselves with the final excision step.
The umkehr map proceeds as follows.
Suppose f : Xn → Y m is a PL map of compact oriented PL stratified pseudomanifolds
such that f−1(Y m−1) ⊂ Xn−1, where Xn−1 and Y m−1 are the respective singular sets of X
and Y . Suppose that D ∈ Ci(Y ;R0) = Ci(Y ;Y m−1). Then by Lemma 5.1, D corresponds to
the homology class [D] ∈ Hi(|D|∪Y m−1, |∂D|∪Y m−1;R) ∼= Hi(|D|, |∂D|∪ (|D|∩Y m−1);R).
Let A = |D|, B = |∂D|, A′ = f−1(A), and B′ = f−1(B). We consider the following
composition of maps, in which all groups have R coefficients.
S−mHi(A ∪ Y m−1, B ∪ Y m−1)(−1)m(m−i)(· ∩ [Y ])−1
∼=- Hm−i(Y − (B ∪ Y m−1), Y − (A ∪ Y m−1))
f ∗- Hm−i(X − (B′ ∪Xn−1), X − (A′ ∪Xn−1))
(−1)n(m−i)(· ∩ [X])∼=
- S−nHi+n−m(A′ ∪Xn−1, B′ ∪Xn−1)
The indicated signed cap products with the respective fundamental classes again represent
the Whitehead-Dold-Goresky-MacPherson duality isomorphism - see [28, Appendix]. We
also incorporate the sign conventions of [25].
If dim(A′ − A′ ∩ Xn−1) ≤ i + n − m and dim(B′ − B′ ∩ Xn−1) ≤ i + n − m −1, then by Lemma 5.1, the image of this composition represents a well-defined chain in
S−nCi+n−m(X,Xn−1;R) = S−nCi+n−m(X;R0). In this case, we say that D is in general po-
sition with respect to f , and we denote the composite map Ci(Y ;R0)→ Ci+n−m(X;R0) by
35
f!. Furthermore, f! is a degree 0 chain map as follows from the same arguments as presented
in the proof of [25, Lemma 7.3].
In particular now, suppose that ∆ : X → X(k) is the diagonal embedding x→ (x, . . . , x).
Then ∆ = R∗ for the unique surjection R : k → 1. For some collection of general perver-
sities P , suppose D ∈ GPk,0. Then by assumption, εD is in stratified general position, so
by the definition of the last section, dim(∆−1(Z(k) ∩ ρ(εD))) = dim(Z ∩ ∆−1(ρ(εD))) ≤dim(Z(k) ∩ ρ(εD)) + dim(Z)(1 − k). In particular, if D is (the shift of) an i-chain and Z
is a regular stratum, we get dim((X −Xn−1) ∩∆−1(ρ(εD))) = dim(∆−1((X −Xn−1)(k) ∩ρ(εD))) ≤ i + n(1 − k), and similarly for ∂D. Thus we have a well-defined ∆! : GP
k,0 →S−nC∗(X,X
n−1;R) ∼= S−nC∗(X;R0), which is the generalized intersection pairing. Further-
more, using analogous computations to the GM perversity case (see [25, Proposition 4.5]),
which extend the sort of computations in our discussion of the Goresky-MacPherson pairing
above, we see that ∆!(D) ∈ S−nI rC∗(X;R0) for any r ≥∑k
i=1 pi.
Note that in the special case where D ∈ GP2,0 and D = S−nx⊗ S−ny, the condition that
ε(D) be in stratified general position with respect to ∆ : X → X ×X says precisely that for
each stratum Z, Z ∩∆−1(|εD|) = dim(Z ∩ |x| ∩ |y|) ≤ dim((Z×Z)∩ (|x|× |y|))−dim(Z) =
dim(|x| ∩ Z) + dim(|y| ∩ Z) − dim(Z), which is precisely the condition that |x| and |y| be
in stratified general position. Similarly, the condition that ε(∂D) be in stratified general
position with respect to ∆ is equivalent to the pairs (|x|, |∂y|) and (|∂x|, |y|) each being in
general position. So, in this special setting, the conditions for ∆! to be well-defined reduce
to those for the Goresky-MacPherson pairing, and arguments as those in the proof of [25,
Proposition 4.9] show that these pairings are equivalent, up to the appropriate index shifts.
It also follows analogously to the arguments of [25, Section 6] that these geometric in-
tersection chain pairings are compatible with the purely sheaf theoretic pairings of Theorem
4.6.
Further results from [25]. Now that we have established the necessary modifications
to the umkehr map and, more generally, understood the role that stratified coefficients
and relative chains play in the generalization of the pairing and general position arguments
to the general perversity case, the remaining major results of [25] go through with little
difficulty. In particular, one can define a category GPn of general perverse chain complexes
consisting of functors from the poset category of general n-perversities (in which p → q
exists uniquely if and only if p(Z) ≤ q(Z) for all singular strata Z) to the category of chain
complexes. Loosely perverse chain complexes are denoted {C?∗} with evaluation at p denoted
by {C?∗}p = C p
∗ . GPn is a symmetric monoidal category with product � obtained by setting
({D?∗}� {E?
∗})r = lim−→p+q≤r
Dp∗ ⊗ E q
∗ .
It follows as by the proof of [25, Theorem 5.3], with some minor modifications, that the
following theorem holds:
Theorem 5.10. For any compact oriented PL stratified pseudomanifold X, the partially-
defined intersection pairing on the perverse chain complex {S−nI?C∗(X,R0)} extends to the
structure of a partial perverse commutative DGA.
36
A partial perverse commutative DGA is what we referred to as a partial restricted com-
mutative DGA in [25, Section 5] with the change that we now allow general perversities.
This wording is also more reflective of the fact that we no longer need restrict to perversities
(or sums of perversities) below t. We refer the reader to [25, Section 5] for the original
definition of this structure and the details of the proof, which readily generalize in light of
the work above.
References
[1] Markus Banagl, The signature of singular spaces and its refinement to generalized ho-
mology theory, To appear in Topology of Stratified Spaces, Mathematical Sciences Re-
search Institute Publications 58, Cambridge University Press, New York, 2010.
[2] , Topological invariants of stratified spaces, Springer Monographs in Mathemat-
ics, Springer-Verlag, New York, 2006.
[3] Markus Banagl, The signature of partially defined local coefficient systems, J. Knot
Theory Ramifications 17 (2008), no. 12, 1455–1481.
[4] Markus Banagl, Sylvain Cappell, and Julius Shaneson, Computing twisted signatures
and L-classes of stratified spaces, Math. Ann. 326 (2003), 589–623.
[5] A.A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Asterisque 100 (1982),
5–171.
[6] Armand Borel, Introduction to middle intersection cohomology and perverse sheaves,
Algebraic groups and their generalizations: classical methods (University Park, PA,
1991), Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994,
pp. 25–52. MR MR1278699 (95h:55006)
[7] A. Borel et. al., Intersection cohomology, Progress in Mathematics, vol. 50, Birkhauser,
Boston, 1984.
[8] Sylvain Cappell, Julius Shaneson, and Shmuel Weinberger, Classes topologiques car-
acteristiques pour les actions de groupes sur les espaces singuliers, C. R. Acad. Sci.
Paris Ser. I Math. 313 (1991), no. 5, 293–295.
[9] Sylvain E. Cappell, Anatoly Libgober, Laurentiu G. Maxim, and Julius L. Shaneson,
Hodge genera of algebraic varieties. II, Math. Ann. 345 (2009), no. 4, 925–972.
[10] Sylvain E. Cappell, Laurentiu G. Maxim, and Julius L. Shaneson, Hodge genera of
algebraic varieties. I, Comm. Pure Appl. Math. 61 (2008), no. 3, 422–449.
[11] Sylvain E. Cappell and Julius L. Shaneson, Singular spaces, characteristic classes, and
intersection homology, Annals of Mathematics 134 (1991), 325–374.
37
[12] G. Carron, L2-cohomology of manifolds with flat ends, Geom. Funct. Anal. 13 (2003),
no. 2, 366–395.
[13] Gilles Carron, Cohomologie L2 des varietes QALE, Preprint, 2005.
[14] , Cohomologie L2 et parabolicite, J. Geom. Anal. 15 (2005), no. 3, 391–404.
[15] Jeff Cheeger and Xianzhe Dai, L2-cohomology of a non-isolated conical singularity and
nonmultiplicatity of signatures, Preprint.
[16] Daniel C. Cohen, Mark Goresky, and Lizhen Ji, On the Kunneth formula for intersection
cohomology, Trans. Amer. Math. Soc. 333 (1992), 63–69.
[17] Xianzhe Dai, Adiabatic limits, nonmultiplicativity of signature, and Leray spectral se-
quence, J. Amer. Math. Soc. 4 (1991), no. 2, 265–321.
[18] Alexandru Dimca, Sheave in topology, Springer-Verlag, Berlin-Heidelberg-New York,
2004.
[19] Greg Friedman, An introduction to intersection homology with general perversity func-
tions, To appear in Topology of Stratified Spaces, Mathematical Sciences Research
Institute Publications 58, Cambridge University Press, New York, 2010.
[20] , Superperverse intersection cohomology: stratification (in)dependence, Math. Z.
252 (2006), 49–70.
[21] , Intersection homology of stratified fibrations and neighborhoods, Adv. Math.
215 (2007), no. 1, 24–65.
[22] , Singular chain intersection homology for traditional and super-perversities,
Trans. Amer. Math. Soc. 359 (2007), 1977–2019.
[23] , Intersection homology and Poincare duality on homotopically stratified spaces,
Geom. Topol. 13 (2009), 2163–2204.
[24] Greg Friedman, Intersection homology Kunneth theorems, Math. Ann. 343 (2009), no. 2,
371–395.
[25] Greg Friedman, On the chain-level intersection pairing for PL pseudomanifolds, Homol-
ogy, Homotopy and Applications 11 (2009), 261–314.
[26] Greg Friedman and Eugenie Hunsicker, Additivity and non-additivity for perverse sig-
natures, preprint; see http://arxiv.org/abs/0911.3915.
[27] Greg Friedman and James McClure, Cup and cap products in intersection (co)homology,
in preparation.
[28] Mark Goresky and Robert MacPherson, Intersection homology theory, Topology 19
(1980), 135–162.
38
[29] , Intersection homology II, Invent. Math. 72 (1983), 77–129.
[30] Mark Goresky and Paul Siegel, Linking pairings on singular spaces, Comment. Math.
Helvetici 58 (1983), 96–110.
[31] Nathan Habegger, From PL to sheaf theory (rock to Bach), Intersection cohomology
(Bern, 1983), Progr. Math., vol. 50, Birkhauser Boston, Boston, MA, 1984, pp. 23–34.
[32] Nathan Habegger and Leslie Saper, Intersection cohomology of cs-spaces and Zeeman’s
filtration, Invent. Math. 105 (1991), 247–272.
[33] A. Haefliger, Introduction to piecewise linear intersection homology, Intersection coho-
mology (Bern, 1983), Progr. Math., vol. 50, Birkhauser Boston, Boston, MA, 1984,
pp. 1–21.
[34] Tamas Hausel, Eugenie Hunsicker, and Rafe Mazzeo, Hodge cohomology of gravitational
instantons, Duke Math. J. 122 (2004), no. 3, 485–548.
[35] Eugenie Hunsicker, Hodge and signature theorems for a family of manifolds with fibre
bundle boundary, Geom. Topol. 11 (2007), 1581–1622.
[36] Eugenie Hunsicker and Rafe Mazzeo, Harmonic forms on manifolds with edges, Int.
Math. Res. Not. (2005), no. 52, 3229–3272.
[37] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathe-
matischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292,
Springer-Verlag, Berlin, 1994.
[38] Henry C. King, Topological invariance of intersection homology without sheaves, Topol-
ogy Appl. 20 (1985), 149–160.
[39] Frances Kirwan and Jonathan Woolf, An introduction to intersection homology theory.
second edition, Chapman & Hall/CRC, Boca Raton, FL, 2006.
[40] Robert MacPherson, Intersection homology and perverse sheaves, Unpublished Collo-
quium Lectures.
[41] Charles Richard Francis Maunder, Algebraic topology, Cambridge University Press,
Cambridge, 1980.
[42] J.E. McClure, On the chain-level intersection pairing for PL manifolds, Geom. Topol.
10 (2006), 1391–1424.
[43] Werner Muller, L2-index theory, eta invariants and values of L-functions, Geometric
and topological invariants of elliptic operators (Brunswick, ME, 1988), Contemp. Math.,
vol. 105, Amer. Math. Soc., Providence, RI, 1990, pp. 145–189.
[44] James R. Munkres, Elements of algebraic topology, Addison-Wesley, Reading, MA, 1984.
39
[45] Leslie Saper, L2-cohomology of locally symmetric spaces. I, Pure Appl. Math. Q. 1
(2005), no. 4, part 3, 889–937.
[46] , L -modules and the conjecture of Rapoport and Goresky-MacPherson, Auto-
morphic forms. I. Asterisque (2005), no. 298, 319–334.
[47] Leslie Saper and Mark Stern, L2-cohomology of arithmetic varieties, Ann. of Math. (2)
132 (1990), no. 1, 1–69.
[48] Martintxo Saralegi-Aranguren, de Rham intersection cohomology for general perversi-
ties, Illinois J. Math. 49 (2005), no. 3, 737–758 (electronic).
[49] P.H. Siegel, Witt spaces: a geometric cycle theory for KO-homology at odd primes,
American J. Math. 110 (1934), 571–92.
40